Drorbn - User contributions [en]

Talk:06-240

2006-12-14T00:11:27Z

Wanmike:

If anyone is interested in typesetting the lectures I think they should follow these Wikipedia guidelines: http://en.wikipedia.org/wiki/WP:MSM

== Modular Arithmetic ==

This was particularly interesting after being introduced to Modular Multiplication tables and seeing some visual patterns with the numbers, such as the in the '1' column where the elements go from 1 to n-1 in Zn and backwards in the 'n-1' column.

After searching around, it seems that people had been able to discover other, more interesting patterns!

Make sure to analyze the tables since they begin from the bottom left corner instead of top left which we saw in class.

http://whistleralley.com/mod/mod25.htm

The following site allows you to see tables up to mod 30.

http://www.cut-the-knot.org/blue/Modulo.shtml

-Richard

Also, notice how in modular multiplication tables for prime numbers, in specific for modulo 5 in the columns and rows for 0 and 5 only 0s appear. The 0s create a sort of frame around a 4x4 square of elements. Specifically all elements within the frame of 0s are between 1 and n-1 and all are non-zero. In the case of the mod 4 table there was a 0 which, as proved in class causes Z4 to fail as a field. There must be something deeper about all those 0s.

== Mistake in the timetable in the main page of 06-240? ==

In the timetable in the main page of 06-240,i.e. http://katlas.math.toronto.edu/drorbn/index.php?title=06-240 , I think there's a mistake about the date of the first test. There, it's written Oct 23th, but we have no class on Oct 23 which is a Monday. I guess the correct date should be Oct 24th as written in the course outline.

-Yanshuai

The dates on the time table are the dates of the Mondays in each week; the header says "week of...". --[[User:Drorbn|Drorbn]] 17:46, 9 October 2006 (EDT)

== TA Office Hours? ==

Do they have any?

06-240/On The Final Exam

2006-12-13T23:30:57Z

Wanmike:

{{06-240/Navigation}}

Our final exam is coming up. It will take place at BN3 - room 3 of the Clara Benson Building, 320 Huron Street (south west of Harbord cross Huron, home of the Faculty of Physical Education and Health) on Wednesday December 13 from 2PM until 5PM. It will consist of 5-6 questions (each may have several parts) on everything that we will have covered in class this semester:
*Fields and vector spaces.
*Spans, independence, replacement and bases.
*Linear transformation, rank, nullity, matrices.
*Row and column reduction and elementary matrices, systems of linear equations.
*Determinants.
*Change of basis and diagonalization.
*Several other "smaller" topics.

As for the style -

*You can expect to be asked to reproduce some proofs that were given in class.
*You can expect some fresh things to prove, though generally not as hard as the previous type of proofs.
*You can expect questions (or parts of questions) that will be identical or nearly identical to questions that were assigned for homework.
*You can expect some calculations (but nothing that will require a calculator).

Basic calculators (not capable of displaying text or sounding speech) will be allowed but will not be necessary. You may wish to bring one nevertheless, as under pressure <math>5+7</math> often comes out to be <math>13</math>.

'''Remember.''' Neatness counts! Organization counts! Language counts! Proofs are best given as short and readable essays; without the English between the formulas one never knows how to interpret those formulas. When you write, say, "<math>v\in V</math>", does it mean "choose <math>v\in V</math>", or "we've just proven that <math>v\in V</math>", or "assume by contradiction that <math>v\in V</math>", or "for every <math>v\in V</math>" or "there exists <math>v\in V</math>"? If you don't say, your reader has no way of knowing. Also remember that long and roundabout solutions of simple problems, full of detours and irrelevant facts, are often an indication that their author didn't quite get the point, even if they are entirely correct. Avoid those!

'''Office hours.''' I ({{Dror}}) will hold two sessions of extended office hours at or near my office (Bahen 6178) before the final:
* On Tuesday December 12th (the day before), 1-4PM.
* On Wednesday December 13th (the day of), 10-12, just for last minute questions.
I would have loved to give more, but long before the final was scheduled I was asked to organize a [[CMS Winter 2006 Session on Knot Homologies|session]] in a Canadian Mathematical Society conference on Saturday through Monday right before our exam. So unfortunately I will be completely unreachable on these three days.

Here is a link to the exam forum: [[06-240/Final Exam Preparation Forum|Final Exam Preparation Forum]]

06-240

2006-12-13T23:30:15Z

Wanmike:

__NOEDITSECTION__
{{06-240/Navigation}}

==Algebra I==
===Department of Mathematics, University of Toronto, Fall 2006===

{{06-240/Crucial Information}}

===Text===
Freidberg, Insel, Spence. Linear Algebra, 4e. New Jersy: Pearson Education Inc, 2003. 

===Famous Quotes About Math and Mind===
* "The problems that exist in the world today cannot be solved by the level of thinking that created them." by Albert Einstein
* "Do not worry about your difficulties in mathematics, I can assure you mine are still greater." by Albert Einstein
* "The mind, once expanded to the dimensions of larger ideas, never returns to its original size." by Oliver Wendell Holmes, Sr.
* "I don't think necessity is the mother of invention. Invention, in my opinion, arises directly from idleness, possibly also from laziness - to save oneself trouble." by Dame Agatha Christie (1890-1976)
* "A mathematician is a machine for turning coffee into theorems." by Alfréd Rényi, colleague of Paul Erdős.

===Further Resources===

* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]
* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].
* [http://www.math.toronto.edu/murnaghan/courses/mat240/index.html Last year's MAT240 web site].
* [http://www.math.toronto.edu/~megumi/MAT240/240.html Two years ago's MAT240 site].
* [http://www.maths.leeds.ac.uk/~khouston/httlam.html "How to Think Like a Mathematician"] by Kevin Houston.
* [http://www.stonehill.edu/compsci/History_Math/math-read.htm "How to Read Mathematics"] by Shai Simonson and Fernando Gouvea.
* A wide variety of free online mathematics texts is [http://www.math.gatech.edu/~cain/textbooks/onlinebooks.html here]. Want more examples? Get them here. Also, for those in 157 I'd suggest looking at Elias Zakon's Analysis 1 text as a supplement for Spivak.

06-240/Final Exam Preparation Forum

2006-12-13T15:48:12Z

Wanmike: /* Exam April 2004 #6(a) */

{{06-240/Navigation}}

If you have questions, ask them here and hopefully someone else will know the answer. (Answering questions will probably help you understand it more).

Since many of us (including I) don't really know how to use Wiki's, I suggest that we keep the formatting simple: I will post a template at the top of this page, and if you want to add something just click on the "edit", copy the template, and insert your question. Order the questions according to section (i.e. solved/unsolved; '''whoever created the question must decide if it is solved, and sort it accordingly'''), with the newest at the top, except for the template question. In general, I wouldn't retype the question if it's from the book because that's tedious and we all have the book.

(By the way, I think you leave a space between lines in the code to make a new line; that is, simply pressing enter once will not make a new line. Also, you can press a button at the top of the editing textbox that lets you put in simple equations.)

==Unsolved Questions==

===Question Template===
Q: Can someone help me prove: "If an [[integer]] <math>n</math> is greater than 2, then <math>a^n + b^n = c^n</math> has no solutions in non-zero integers <math>a</math>, <math>b</math>, and <math>c</math>."? I had the answer in my head at one point, but the margins of the piece of paper I was working with was too small to fit it.

===Rank of Matrices===
Q: Prove that if rankA = 0 (for A with dimension m x n), then A is the zero matrix.
(Question is found on p 166 (#3))
Did anyone use transformations in this?
My proof relies on rank(L_A) = 0 (left-multiplication transformation)implying L_A is the zero transformation.
Is there an easier way?

A: Depends on what you're allowed to assume. If you can use that the rank is equal to the number of linearly independent columns, then clearly all of the columns must be zero (otherwise you have at least one linearly independent vector).

===Determinants===
Q : If we can make same matrix with 2n-1 times of row swaps, what does it mean ? Does it mean that determinant is 0 ?

R: Is this related to a question somewhere?

===Exam April/May 2006 #4===
Q: Suppose that A, B Є Mmxn(F), and rank(A) = rank (B). Prove that there exist invertible matrices P Є Mmxm(F) and Q Є Mnxn(F) such that B = PAQ.

A(partial): Here is a sketch. If you rref A and B by applying a series of elementary row operation matricies, they will both look similar. That is, they will have a section of 1's and 0's (each 1 is the only number in its column) and then a section of "remaining stuff", and these sections will be the same "size" because their ranks are the same. Then, using the elementary column matrix operations, you can essentially modify the "remaining stuff" as much as you like, by adding multiples of the "nice" columns (with single-1's). These row and column operations can then be grouped nicely and set to be equal to P and Q, which are invertible because products of elementary matricies are invertible.

I know this is very rough, but even if I did have a full answer I wouldn't now how to typeset it.

===Complex Numbers===
Q: If 'C' is used in the context of a vector space (as in "define T:C->C"), then should we consider C to be the vector space of C over the field C, or instead C over the field R?

===Readings?===
Q: Are we expected to section 5.2 of the textbook? Although the Assignments tell us to read it, we didn't do any questions, or cover it in class.

A: I highly doubt it; hopefully someone will ask Prof. Bar-Natan tomorrow and post the answer here. There were a few other chapters that had sections we never really talked about either (some applications). Addendum: I second this request for a slight narrowing of what the relevant readings are--for instance, can we be more efficient in our reading of chapter 4 somehow?

R: I think that if you want to cut down on Chapter 4, then skipping applications of area (discussed very briefly in class) and determinants of order 2 is the most you can do.

R: What about 4.5, the axiomatic details. It discusses how the determinant is uniquely defined by the three axiomatic properties, but I don't think we did that in class.

==Solved Questions==

===Question Template===
Q: How many ways are there to get to the nth stair, if at each step you can move either one or two squares up?

A: This question can be easily modeled by the Fibonacci numbers, with the nth number being the ways to get to the nth stair. This is because, to get to the nth stair, you can come only from the n-1th or the n-2th. This is exactly how the Fibonacci numbers are defined; the proof is simple by induction.

===Sec 3.2 Ex. 19===
Q: "Let A be an m x n matrix with rank m and B be an n x p matrix with rank n. Determine the rank of AB. Justify your answer." I know how to find that the rank can't be more than m (not much of an accomplishment), but I can't finish it.

A: According to Theorem 3.7(a),(c)&(d)(p.159), I would say rank(AB) <math>\le </math>min(m, n).

R: Can we not get any more specific than that?

A: "Let <math>L_A, L_B, L_{AB} </math> have their usual meanings. Then <math>L_B : F^p -> F^n </math> is onto. Then we get <math> R(L_{AB}) = R(L_A L_B) = L_A L_B (F^p) = L_A (F^n) = R(L_A) </math>, i.e. <math>rank(L_{AB}) = rank(L_A) = m</math>."

===Sec. 3.2 Ex. 21===
Q: "Let A be an m x n matrix with rank m. Prove that there exists an n x m matrix B such that AB=<math>I_m</math>".

A: "Take any n x m matrix B with rank n. By exercise 19 in the same section rank AB = rank A = m, hence AB is invertible. Let M be the inverse of AB, then (AB)M = A(BM) = I, i.e. BM is the desired matrix."

===Sec. 1.3 Thm 1.3 Proof===
Q: In the first paragraph of the proof, it says "But also x + 0 = x , and thus 0'=0." How do we know 0 (that is 0 of V) even exists in W? I understand that we know ''some'' zero exists (0'), but not why ''the'' zero (0) exists.

A: x is in W as well in V. Thus, x + 0 = x (VS 3).

Reply: Oh I see... now it looks so obvious =/. Thanks.

===Exam April/May 2006 #3(b)===
Q: Let T : M3x2(C) -> M2x3(C) be defined as follows. Given A Є M3x2(C), let B be the matrix obtained from A by adding i times the second row of A to the third row of A. Let T(A) = Bt, where Bt is the transpose of B. (Note: Here, i is a complex number such that i2 = -1.) Determine whether the linear transformation T is invertible.

Totally lost on this question :/ Please show some example matrix and how it is transformed as the question asks if possible. I want to see what actually happens to the elements in the matrix rather than the answer (think that would be more important)

A(Matrix Elements):
This is my interpretation:

A = <math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}</math> where <math>a_{ij} \in C</math>, then B = <math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\ia_{21}+a_{31}&ia_{22}+a_{32}\end{pmatrix}</math>.

Therefore, T(A) = Bt is T<math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}</math>=<math>\begin{pmatrix}a_{11}&a_{21}&ia_{21}+a_{31}\\a_{12}&a_{22}&ia_{22}+a_{32}\end{pmatrix}</math>

R: Thx alot, the matricies are really helpful :)

===Sec. 2.4 Lemma p. 101===
Q: In the proof of the lemma, the second line, we have "T(beta) spans R(T) = W". How do we know that R(T) = W? This would be true if dim V = dim W, because then T would be onto, but we can't assume what we're trying to prove.

A: The first line of the Lemma states, "Let T be an '''invertible''' linear trans..." So, T is onto(and 1-1), thus "T(beta) spans R(T) = W".

R: Yes. My trouble was with the fact that invertibility implies onto-ness. I thought that if we had <math> T:P_2 (R)->P_6(R) </math>, and T(f) = xf, then T would still be invertible since you can 'recover' the f if you were given xf. I guess it makes more sense to not call T invertible in this case, because <math>T^{-1}</math> is technically only one-to-one over the range of T.

R: T has to be both Onto and 1-1 so that it's invertible. In your example, some of the <math> f \in P_6(R)</math> will not be 'recovered' because they weren't mapped from <math> P_2(R)</math>. Furthermore, T-1 has to map the whole vector space W back to V(as defined on p.99) but not the range only. In other words, if T is 1-1 only, <math>T^{-1} \circ T(v) = v, \forall v\in V</math> but <math>T \circ T^{-1}(w) \neq w, \exists w\in W</math>, because some T-1(w) are not defined.

R: That nicely rigorizes what I was thinking, and I'm convinced. Thanks.

===Exam April/May 2006 #7===
Q: Let T : V -> V and U : V -> V be linear operators on a finite-dimensional vector space V. Assume that U is invertible and T is diagonalizable. Prove that the linear operator UTU-1 = U o T o U-1.

I dont know where or how to start this question ><.

A: I think we need to prove that UTU-1 is diagonalizable instead of proving UTU-1 = U o T o U-1.

I started by letting A = UTU-1, then multiplying U-1 and U to the both sides, we get U-1AU = U-1UTU-1U iff U-1AU = T. Since T is diagonalizable, therefore there exists an invertible matrix Q s.t Q-1TQ = D, where D is a diagonal matrix. Therefore, Q-1(U-1AU)Q = D iff (UQ)-1A(UQ) = D (because U and Q invertible, Q-1U-1 = (UQ)-1), it follows that A is diagonalizable.

===Exam April 2004 #6(a)===
Q: Suppose A is an invertible matrix for which the sum of entries of each row is a scalar <math>\lambda</math>. Show that the sum of entries of each row of A-1 is <math>1/\lambda</math>. (Hint: find an eigenvector for A with eigenvalue <math>\lambda</math>.)
If A is a diagonal matrix, then it's obvious that the sum of entries of each row is <math>\lambda</math> and the sum of entries of each row of A-1 is <math>1/\lambda</math>. I was stuck with a more general invertible matrix.

A: Following the hint, you can see that an eigenvector corresponding to <math>\lambda</math> is (1, 1, 1, ...)*. Therefore <math> Av=\lambda v</math>, and rearranging you get <math> A^{-1}v=1/\lambda v</math>. Using the same logic as before, you can show that since this <math>\lambda</math> corresponds to a homogeneous system of equations with the same eigenvector v = (1, 1, 1, ...), the sum of each row is equal to <math>1/\lambda</math>.

*Just to elaborate on the first part, you are looking for a vector <math> v = (x_1, x_2, x_3, ...) </math> so that <math> A-\lambda I = 0</math>. This corresponds to the system:
<math>\begin{pmatrix}(a_{11}-\lambda)x_1&a_{12}x_2&a_{13}x_3&...\\a_{21}x_1&(a_{22}-\lambda)x_2&a_{23}x_3&...\\
a_{11}x_1&a_{12}x_1&(a_{13}-\lambda)x_3&...\end{pmatrix}</math>,
and so in each row you can see that <math>x_1=1, x_2=1, x_3=1</math> works because then all the a's in each row add up to <math>\lambda</math>.

*Also, does anyone know how to do part (b) of that question? My guess is to make one subspace {0}, the second (t,0,0) and the third (0,r,s) for all t,r,s,. Does that look okay?

R: Thanks. I think the subspaces are {0}, {(t,0,0)} and {(0,s,0)} so that <math>R^3 \neq W_1 \oplus W_2 \oplus W_3</math>.

R: We need them to add up to <math>R_3</math> though. Anyway, hopefully we won't need to know about direct sums.

06-240/Final Exam Preparation Forum

2006-12-13T15:46:41Z

Wanmike:

06-240/Final Exam Preparation Forum

2006-12-13T00:55:12Z

Wanmike:

{{06-240/Navigation}}

If you have questions, ask them here and hopefully someone else will know the answer. (Answering questions will probably help you understand it more).

Since many of us (including I) don't really know how to use Wiki's, I suggest that we keep the formatting simple: I will post a template at the top of this page, and if you want to add something just click on the "edit", copy the template, and insert your question. Order the questions according to section (i.e. solved/unsolved; '''whoever created the question must decide if it is solved, and sort it accordingly'''), with the newest at the top, except for the template question. In general, I wouldn't retype the question if it's from the book because that's tedious and we all have the book.

(By the way, I think you leave a space between lines in the code to make a new line; that is, simply pressing enter once will not make a new line. Also, you can press a button at the top of the editing textbox that lets you put in simple equations.)

==Unsolved Questions==

===Question Template===
Q: Can someone help me prove: "If an [[integer]] <math>n</math> is greater than 2, then <math>a^n + b^n = c^n</math> has no solutions in non-zero integers <math>a</math>, <math>b</math>, and <math>c</math>."? I had the answer in my head at one point, but the margins of the piece of paper I was working with was too small to fit it.

===Rank of Matrices===
Q: Prove that if rankA = 0 (for A with dimension m x n), then A is the zero matrix.
(Question is found on p 166 (#3))
Did anyone use transformations in this?
My proof relies on rank(L_A) = 0 (left-multiplication transformation)implying L_A is the zero transformation.
Is there an easier way?

A: Depends on what you're allowed to assume. If you can use that the rank is equal to the number of linearly independent columns, then clearly all of the columns must be zero (otherwise you have at least one linearly independent vector).

===Determinants===
Q : If we can make same matrix with 2n-1 times of row swaps, what does it mean ? Does it mean that determinant is 0 ?

R: Is this related to a question somewhere?

===Sec 3.2 Ex. 19===
Q: "Let A be an m x n matrix with rank m and B be an n x p matrix with rank n. Determine the rank of AB. Justify your answer." I know how to find that the rank can't be more than m (not much of an accomplishment), but I can't finish it.

A: According to Theorem 3.7(a),(c)&(d)(p.159), I would say rank(AB) <math>\le </math>min(m, n).

R: Can we not get any more specific than that?

===Sec. 3.2 Ex. 21===
Q: "Let A be an m x n matrix with rank m. Prove that there exists an n x m matrix B such that AB=<math>I_m</math>".

===Exam April/May 2006 #4===
Q: Suppose that A, B Є Mmxn(F), and rank(A) = rank (B). Prove that there exist invertible matrices P Є Mmxm(F) and Q Є Mnxn(F) such that B = PAQ.

A(partial): Here is a sketch. If you rref A and B by applying a series of elementary row operation matricies, they will both look similar. That is, they will have a section of 1's and 0's (each 1 is the only number in its column) and then a section of "remaining stuff", and these sections will be the same "size" because their ranks are the same. Then, using the elementary column matrix operations, you can essentially modify the "remaining stuff" as much as you like, by adding multiples of the "nice" columns (with single-1's). These row and column operations can then be grouped nicely and set to be equal to P and Q, which are invertible because products of elementary matricies are invertible.

I know this is very rough, but even if I did have a full answer I wouldn't now how to typeset it.

===Complex Numbers===
Q: If 'C' is used in the context of a vector space (as in "define T:C->C"), then should we consider C to be the vector space of C over the field C, or instead C over the field R?

===Readings?===
Q: Are we expected to section 5.2 of the textbook? Although the Assignments tell us to read it, we didn't do any questions, or cover it in class.

A: I highly doubt it; hopefully someone will ask Prof. Bar-Natan tomorrow and post the answer here. There were a few other chapters that had sections we never really talked about either (some applications). Addendum: I second this request for a slight narrowing of what the relevant readings are--for instance, can we be more efficient in our reading of chapter 4 somehow?

R: I think that if you want to cut down on Chapter 4, then skipping applications of area (discussed very briefly in class) and determinants of order 2 is the most you can do.

R: What about 4.5, the axiomatic details. It discusses how the determinant is uniquely defined by the three axiomatic properties, but I don't think we did that in class.

===Exam April 2004 #6(a)===
Q: Suppose A is an invertible matrix for which the sum of entries of each row is a scalar <math>\lambda</math>. Show that the sum of entries of each row of A-1 is <math>1/\lambda</math>. (Hint: find an eigenvector for A with eigenvalue <math>\lambda</math>.)
If A is a diagonal matrix, then it's obvious that the sum of entries of each row is <math>\lambda</math> and the sum of entries of each row of A-1 is <math>1/\lambda</math>. I was stuck with a more general invertible matrix.

A: Following the hint, you can see that an eigenvector corresponding to <math>\lambda</math> is (1, 1, 1, ...)*. Therefore <math> Av=\lambda v</math>, and rearranging you get <math> A^{-1}v=1/\lambda v</math>. Using the same logic as before, you can show that since this <math>\lambda</math> corresponds to a homogeneous system of equations with the same eigenvector v = (1, 1, 1, ...), the sum of each row is equal to <math>1/\lambda</math>.

*Just to elaborate on the first part, you are looking for a vector <math> v = (x_1, x_2, x_3, ...) </math> so that <math> A-\lambda I = 0</math>. This corresponds to the system:
<math>\begin{pmatrix}(a_{11}-\lambda)x_1&a_{12}x_2&a_{13}x_3&...\\a_{21}x_1&(a_{22}-\lambda)x_2&a_{23}x_3&...\\
a_{11}x_1&a_{12}x_1&(a_{13}-\lambda)x_3&...\end{pmatrix}</math>,
and so in each row you can see that <math>x_1=1, x_2=1, x_3=3</math> works because then all the a's in each row add up to <math>\lambda</math>.

*Also, does anyone know how to do part (b) of that question? My guess is to make one subspace {0}, the second (t,0,0) and the third (0,r,s) for all t,r,s,. Does that look okay?

==Solved Questions==

===Question Template===
Q: How many ways are there to get to the nth stair, if at each step you can move either one or two squares up?

A: This question can be easily modeled by the Fibonacci numbers, with the nth number being the ways to get to the nth stair. This is because, to get to the nth stair, you can come only from the n-1th or the n-2th. This is exactly how the Fibonacci numbers are defined; the proof is simple by induction.

===Sec. 1.3 Thm 1.3 Proof===
Q: In the first paragraph of the proof, it says "But also x + 0 = x , and thus 0'=0." How do we know 0 (that is 0 of V) even exists in W? I understand that we know ''some'' zero exists (0'), but not why ''the'' zero (0) exists.

A: x is in W as well in V. Thus, x + 0 = x (VS 3).

Reply: Oh I see... now it looks so obvious =/. Thanks.

===Exam April/May 2006 #3(b)===
Q: Let T : M3x2(C) -> M2x3(C) be defined as follows. Given A Є M3x2(C), let B be the matrix obtained from A by adding i times the second row of A to the third row of A. Let T(A) = Bt, where Bt is the transpose of B. (Note: Here, i is a complex number such that i2 = -1.) Determine whether the linear transformation T is invertible.

Totally lost on this question :/ Please show some example matrix and how it is transformed as the question asks if possible. I want to see what actually happens to the elements in the matrix rather than the answer (think that would be more important)

A(Matrix Elements):
This is my interpretation:

A = <math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}</math> where <math>a_{ij} \in C</math>, then B = <math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\ia_{21}+a_{31}&ia_{22}+a_{32}\end{pmatrix}</math>.

Therefore, T(A) = Bt is T<math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}</math>=<math>\begin{pmatrix}a_{11}&a_{21}&ia_{21}+a_{31}\\a_{12}&a_{22}&ia_{22}+a_{32}\end{pmatrix}</math>

R: Thx alot, the matricies are really helpful :)

===Sec. 2.4 Lemma p. 101===
Q: In the proof of the lemma, the second line, we have "T(beta) spans R(T) = W". How do we know that R(T) = W? This would be true if dim V = dim W, because then T would be onto, but we can't assume what we're trying to prove.

A: The first line of the Lemma states, "Let T be an '''invertible''' linear trans..." So, T is onto(and 1-1), thus "T(beta) spans R(T) = W".

R: Yes. My trouble was with the fact that invertibility implies onto-ness. I thought that if we had <math> T:P_2 (R)->P_6(R) </math>, and T(f) = xf, then T would still be invertible since you can 'recover' the f if you were given xf. I guess it makes more sense to not call T invertible in this case, because <math>T^{-1}</math> is technically only one-to-one over the range of T.

R: T has to be both Onto and 1-1 so that it's invertible. In your example, some of the <math> f \in P_6(R)</math> will not be 'recovered' because they weren't mapped from <math> P_2(R)</math>. Furthermore, T-1 has to map the whole vector space W back to V(as defined on p.99) but not the range only. In other words, if T is 1-1 only, <math>T^{-1} \circ T(v) = v, \forall v\in V</math> but <math>T \circ T^{-1}(w) \neq w, \exists w\in W</math>, because some T-1(w) are not defined.

R: That nicely rigorizes what I was thinking, and I'm convinced. Thanks.

===Exam April/May 2006 #7===
Q: Let T : V -> V and U : V -> V be linear operators on a finite-dimensional vector space V. Assume that U is invertible and T is diagonalizable. Prove that the linear operator UTU-1 = U o T o U-1.

I dont know where or how to start this question ><.

A: I think we need to prove that UTU-1 is diagonalizable instead of proving UTU-1 = U o T o U-1.

I started by letting A = UTU-1, then multiplying U-1 and U to the both sides, we get U-1AU = U-1UTU-1U iff U-1AU = T. Since T is diagonalizable, therefore there exists an invertible matrix Q s.t Q-1TQ = D, where D is a diagonal matrix. Therefore, Q-1(U-1AU)Q = D iff (UQ)-1A(UQ) = D (because U and Q invertible, Q-1U-1 = (UQ)-1), it follows that A is diagonalizable.

06-240/Final Exam Preparation Forum

2006-12-13T00:32:07Z

Wanmike: /* Exam April 2004 #6(a) */

{{06-240/Navigation}}

If you have questions, ask them here and hopefully someone else will know the answer. (Answering questions will probably help you understand it more).

Since many of us (including I) don't really know how to use Wiki's, I suggest that we keep the formatting simple: I will post a template at the top of this page, and if you want to add something just click on the "edit", copy the template, and insert your question. Order the questions according to section (i.e. solved/unsolved; '''whoever created the question must decide if it is solved'''), with the newest at the top, except for the template question. In general, I wouldn't retype the question if it's from the book because that's tedious and we all have the book.

(By the way, I think you leave a space between lines in the code to make a new line; that is, simply pressing enter once will not make a new line. Also, you can press a button at the top of the editing textbox that lets you put in simple equations.)

==Unsolved Questions==

===Question Template===
Q: Can someone help me prove: "If an [[integer]] <math>n</math> is greater than 2, then <math>a^n + b^n = c^n</math> has no solutions in non-zero integers <math>a</math>, <math>b</math>, and <math>c</math>."? I had the answer in my head at one point, but the margins of the piece of paper I was working with was too small to fit it.

===Rank of Matrices===
Q: Prove that if rankA = 0 (for A with dimension m x n), then A is the zero matrix.
(Question is found on p 166 (#3))
Did anyone use transformations in this?
My proof relies on rank(L_A) = 0 (left-multiplication transformation)implying L_A is the zero transformation.
Is there an easier way?

A: Depends on what you're allowed to assume. If you can use that the rank is equal to the number of linearly independent columns, then clearly all of the columns must be zero (otherwise you have at least one linearly independent vector).

===Determinants===
Q : If we can make same matrix with 2n-1 times of row swaps, what does it mean ? Does it mean that determinant is 0 ?

R: Is this related to a question somewhere?

===Sec 3.2 Ex. 19===
Q: "Let A be an m x n matrix with rank m and B be an n x p matrix with rank n. Determine the rank of AB. Justify your answer." I know how to find that the rank can't be more than m (not much of an accomplishment), but I can't finish it.

A: According to Theorem 3.7(a),(c)&(d)(p.159), I would say rank(AB) <math>\le </math>min(m, n).

R: Can we not get any more specific than that?

===Sec. 3.2 Ex. 21===
Q: "Let A be an m x n matrix with rank m. Prove that there exists an n x m matrix B such that AB=<math>I_m</math>".

===Exam April/May 2006 #4===
Q: Suppose that A, B Є Mmxn(F), and rank(A) = rank (B). Prove that there exist invertible matrices P Є Mmxm(F) and Q Є Mnxn(F) such that B = PAQ.

A(partial): Here is a sketch. If you rref A and B by applying a series of elementary row operation matricies, they will both look similar. That is, they will have a section of 1's and 0's (each 1 is the only number in its column) and then a section of "remaining stuff", and these sections will be the same "size" because their ranks are the same. Then, using the elementary column matrix operations, you can essentially modify the "remaining stuff" as much as you like, by adding multiples of the "nice" columns (with single-1's). These row and column operations can then be grouped nicely and set to be equal to P and Q, which are invertible because products of elementary matricies are invertible.

I know this is very rough, but even if I did have a full answer I wouldn't now how to typeset it.

===Complex Numbers===
Q: If 'C' is used in the context of a vector space (as in "define T:C->C"), then should we consider C to be the vector space of C over the field C, or instead C over the field R?

===Readings?===
Q: Are we expected to section 5.2 of the textbook? Although the Assignments tell us to read it, we didn't do any questions, or cover it in class.

A: I highly doubt it; hopefully someone will ask Prof. Bar-Natan tomorrow and post the answer here. There were a few other chapters that had sections we never really talked about either (some applications). Addendum: I second this request for a slight narrowing of what the relevant readings are--for instance, can we be more efficient in our reading of chapter 4 somehow?

R: I think that if you want to cut down on Chapter 4, then skipping applications of area (discussed very briefly in class) and determinants of order 2 is the most you can do.

R: What about 4.5, the axiomatic details. It discusses how the determinant is uniquely defined by the three axiomatic properties, but I don't think we did that in class.

===Exam April 2004 #6(a)===
Q: Suppose A is an invertible matrix for which the sum of entries of each row is a scalar <math>\lambda</math>. Show that the sum of entries of each row of A-1 is <math>1/\lambda</math>. (Hint: find an eigenvector for A with eigenvalue <math>\lambda</math>.)
If A is a diagonal matrix, then it's obvious that the sum of entries of each row is <math>\lambda</math> and the sum of entries of each row of A-1 is <math>1/\lambda</math>. I was stuck with a more general invertible matrix.

A: Following the hint, you can see that an eigenvector corresponding to <math>\lambda</math> is (1, 1, 1, ...)*. Therefore <math> Av=\lambda v</math>, and rearranging you get <math> A^{-1}v=1/\lambda v</math>. Using the same logic as before, you can show that since this <math>\lambda</math> corresponds to a homogeneous system of equations with the same eigenvector v = (1, 1, 1, ...), the sum of each row is equal to <math>1/\lambda</math>.

*Just to elaborate on the first part, you are looking for a vector <math> v = (x_1, x_2, x_3, ...) </math> so that <math> A-\lambda I = 0</math>. This corresponds to the system:
<math>\begin{pmatrix}(a_{11}-\lambda)x_1&a_{12}x_2&a_{13}x_3&...\\a_{21}x_1&(a_{22}-\lambda)x_2&a_{23}x_3&...\\
a_{11}x_1&a_{12}x_1&(a_{13}-\lambda)x_3&...\end{pmatrix}</math>,
and so in each row you can see that <math>x_1=1, x_2=1, x_3=3</math> works because then all the a's in each row add up to <math>\lambda</math>.

*Also, does anyone know how to do part (b) of that question? My guess is to make one subspace {0}, the second (t,0,0) and the third (0,r,s) for all t,r,s,. Does that look okay?

==Solved Questions==

===Question Template===
Q: How many ways are there to get to the nth stair, if at each step you can move either one or two squares up?

A: This question can be easily modeled by the Fibonacci numbers, with the nth number being the ways to get to the nth stair. This is because, to get to the nth stair, you can come only from the n-1th or the n-2th. This is exactly how the Fibonacci numbers are defined; the proof is simple by induction.

===Sec. 1.3 Thm 1.3 Proof===
Q: In the first paragraph of the proof, it says "But also x + 0 = x , and thus 0'=0." How do we know 0 (that is 0 of V) even exists in W? I understand that we know ''some'' zero exists (0'), but not why ''the'' zero (0) exists.

A: x is in W as well in V. Thus, x + 0 = x (VS 3).

Reply: Oh I see... now it looks so obvious =/. Thanks.

===Exam April/May 2006 #3(b)===
Q: Let T : M3x2(C) -> M2x3(C) be defined as follows. Given A Є M3x2(C), let B be the matrix obtained from A by adding i times the second row of A to the third row of A. Let T(A) = Bt, where Bt is the transpose of B. (Note: Here, i is a complex number such that i2 = -1.) Determine whether the linear transformation T is invertible.

Totally lost on this question :/ Please show some example matrix and how it is transformed as the question asks if possible. I want to see what actually happens to the elements in the matrix rather than the answer (think that would be more important)

A(Matrix Elements):
This is my interpretation:

A = <math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}</math> where <math>a_{ij} \in C</math>, then B = <math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\ia_{21}+a_{31}&ia_{22}+a_{32}\end{pmatrix}</math>.

Therefore, T(A) = Bt is T<math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}</math>=<math>\begin{pmatrix}a_{11}&a_{21}&ia_{21}+a_{31}\\a_{12}&a_{22}&ia_{22}+a_{32}\end{pmatrix}</math>

R: Thx alot, the matricies are really helpful :)

===Sec. 2.4 Lemma p. 101===
Q: In the proof of the lemma, the second line, we have "T(beta) spans R(T) = W". How do we know that R(T) = W? This would be true if dim V = dim W, because then T would be onto, but we can't assume what we're trying to prove.

A: The first line of the Lemma states, "Let T be an '''invertible''' linear trans..." So, T is onto(and 1-1), thus "T(beta) spans R(T) = W".

R: Yes. My trouble was with the fact that invertibility implies onto-ness. I thought that if we had <math> T:P_2 (R)->P_6(R) </math>, and T(f) = xf, then T would still be invertible since you can 'recover' the f if you were given xf. I guess it makes more sense to not call T invertible in this case, because <math>T^{-1}</math> is technically only one-to-one over the range of T.

R: T has to be both Onto and 1-1 so that it's invertible. In your example, some of the <math> f \in P_6(R)</math> will not be 'recovered' because they weren't mapped from <math> P_2(R)</math>. Furthermore, T-1 has to map the whole vector space W back to V(as defined on p.99) but not the range only. In other words, if T is 1-1 only, <math>T^{-1} \circ T(v) = v, \forall v\in V</math> but <math>T \circ T^{-1}(w) \neq w, \exists w\in W</math>, because some T-1(w) are not defined.

R: That nicely rigorizes what I was thinking, and I'm convinced. Thanks.

===Exam April/May 2006 #7===
Q: Let T : V -> V and U : V -> V be linear operators on a finite-dimensional vector space V. Assume that U is invertible and T is diagonalizable. Prove that the linear operator UTU-1 = U o T o U-1.

I dont know where or how to start this question ><.

A: I think we need to prove that UTU-1 is diagonalizable instead of proving UTU-1 = U o T o U-1.

I started by letting A = UTU-1, then multiplying U-1 and U to the both sides, we get U-1AU = U-1UTU-1U iff U-1AU = T. Since T is diagonalizable, therefore there exists an invertible matrix Q s.t Q-1TQ = D, where D is a diagonal matrix. Therefore, Q-1(U-1AU)Q = D iff (UQ)-1A(UQ) = D (because U and Q invertible, Q-1U-1 = (UQ)-1), it follows that A is diagonalizable.

06-240/Final Exam Preparation Forum

2006-12-13T00:31:00Z

Wanmike: /* Exam April 2004 #6(a) */

{{06-240/Navigation}}

If you have questions, ask them here and hopefully someone else will know the answer. (Answering questions will probably help you understand it more).

Since many of us (including I) don't really know how to use Wiki's, I suggest that we keep the formatting simple: I will post a template at the top of this page, and if you want to add something just click on the "edit", copy the template, and insert your question. Order the questions according to section (i.e. solved/unsolved; '''whoever created the question must decide if it is solved'''), with the newest at the top, except for the template question. In general, I wouldn't retype the question if it's from the book because that's tedious and we all have the book.

(By the way, I think you leave a space between lines in the code to make a new line; that is, simply pressing enter once will not make a new line. Also, you can press a button at the top of the editing textbox that lets you put in simple equations.)

==Unsolved Questions==

===Question Template===
Q: Can someone help me prove: "If an [[integer]] <math>n</math> is greater than 2, then <math>a^n + b^n = c^n</math> has no solutions in non-zero integers <math>a</math>, <math>b</math>, and <math>c</math>."? I had the answer in my head at one point, but the margins of the piece of paper I was working with was too small to fit it.

===Rank of Matrices===
Q: Prove that if rankA = 0 (for A with dimension m x n), then A is the zero matrix.
(Question is found on p 166 (#3))
Did anyone use transformations in this?
My proof relies on rank(L_A) = 0 (left-multiplication transformation)implying L_A is the zero transformation.
Is there an easier way?

A: Depends on what you're allowed to assume. If you can use that the rank is equal to the number of linearly independent columns, then clearly all of the columns must be zero (otherwise you have at least one linearly independent vector).

===Determinants===
Q : If we can make same matrix with 2n-1 times of row swaps, what does it mean ? Does it mean that determinant is 0 ?

R: Is this related to a question somewhere?

===Sec 3.2 Ex. 19===
Q: "Let A be an m x n matrix with rank m and B be an n x p matrix with rank n. Determine the rank of AB. Justify your answer." I know how to find that the rank can't be more than m (not much of an accomplishment), but I can't finish it.

A: According to Theorem 3.7(a),(c)&(d)(p.159), I would say rank(AB) <math>\le </math>min(m, n).

R: Can we not get any more specific than that?

===Sec. 3.2 Ex. 21===
Q: "Let A be an m x n matrix with rank m. Prove that there exists an n x m matrix B such that AB=<math>I_m</math>".

===Exam April/May 2006 #4===
Q: Suppose that A, B Є Mmxn(F), and rank(A) = rank (B). Prove that there exist invertible matrices P Є Mmxm(F) and Q Є Mnxn(F) such that B = PAQ.

A(partial): Here is a sketch. If you rref A and B by applying a series of elementary row operation matricies, they will both look similar. That is, they will have a section of 1's and 0's (each 1 is the only number in its column) and then a section of "remaining stuff", and these sections will be the same "size" because their ranks are the same. Then, using the elementary column matrix operations, you can essentially modify the "remaining stuff" as much as you like, by adding multiples of the "nice" columns (with single-1's). These row and column operations can then be grouped nicely and set to be equal to P and Q, which are invertible because products of elementary matricies are invertible.

I know this is very rough, but even if I did have a full answer I wouldn't now how to typeset it.

===Complex Numbers===
Q: If 'C' is used in the context of a vector space (as in "define T:C->C"), then should we consider C to be the vector space of C over the field C, or instead C over the field R?

===Readings?===
Q: Are we expected to section 5.2 of the textbook? Although the Assignments tell us to read it, we didn't do any questions, or cover it in class.

A: I highly doubt it; hopefully someone will ask Prof. Bar-Natan tomorrow and post the answer here. There were a few other chapters that had sections we never really talked about either (some applications). Addendum: I second this request for a slight narrowing of what the relevant readings are--for instance, can we be more efficient in our reading of chapter 4 somehow?

R: I think that if you want to cut down on Chapter 4, then skipping applications of area (discussed very briefly in class) and determinants of order 2 is the most you can do.

R: What about 4.5, the axiomatic details. It discusses how the determinant is uniquely defined by the three axiomatic properties, but I don't think we did that in class.

===Exam April 2004 #6(a)===
Q: Suppose A is an invertible matrix for which the sum of entries of each row is a scalar <math>\lambda</math>. Show that the sum of entries of each row of A-1 is <math>1/\lambda</math>. (Hint: find an eigenvector for A with eigenvalue <math>\lambda</math>.)
If A is a diagonal matrix, then it's obvious that the sum of entries of each row is <math>\lambda</math> and the sum of entries of each row of A-1 is <math>1/\lambda</math>. I was stuck with a more general invertible matrix.

A: Following the hint, you can see that an eigenvector corresponding to <math>\lambda</math> is (1, 1, 1, ...)*. Therefore <math> Av=\lambda v</math>, and rearranging you get <math> A^{-1}v=1/\lambda v</math>. Using the same logic as before, you can show that since this <math>\lambda</math> corresponds to a homogeneous system of equations with the same eigenvector v = (1, 1, 1, ...), the sum of each row is equal to <math>1/\lambda</math>.

*Just to elaborate on the first part, you are looking for a vector <math> v = (x_1, x_2, x_3, ...) </math> so that <math> A-\lambda I = 0</math>. This corresponds to the system:
<math>\begin{pmatrix}(a_{11}-\lambda)x_1&a_{12}x_2&a_{13}x_3&...\\a_{21}x_1&(a_{22}-\lambda)x_2&a_{23}x_3&...\\
a_{11}x_1&a_{12}x_1&(a_{13}-\lambda)x_3&...\end{pmatrix}</math>,
and so in each row you can see that <math>x_1=1, x_2=1, x_3=3</math> works because then all the a's in each row add up to <math>\lambda</math>.

*Also, does anyone know how to do part (b) of that question?

==Solved Questions==

===Question Template===
Q: How many ways are there to get to the nth stair, if at each step you can move either one or two squares up?

A: This question can be easily modeled by the Fibonacci numbers, with the nth number being the ways to get to the nth stair. This is because, to get to the nth stair, you can come only from the n-1th or the n-2th. This is exactly how the Fibonacci numbers are defined; the proof is simple by induction.

===Sec. 1.3 Thm 1.3 Proof===
Q: In the first paragraph of the proof, it says "But also x + 0 = x , and thus 0'=0." How do we know 0 (that is 0 of V) even exists in W? I understand that we know ''some'' zero exists (0'), but not why ''the'' zero (0) exists.

A: x is in W as well in V. Thus, x + 0 = x (VS 3).

Reply: Oh I see... now it looks so obvious =/. Thanks.

===Exam April/May 2006 #3(b)===
Q: Let T : M3x2(C) -> M2x3(C) be defined as follows. Given A Є M3x2(C), let B be the matrix obtained from A by adding i times the second row of A to the third row of A. Let T(A) = Bt, where Bt is the transpose of B. (Note: Here, i is a complex number such that i2 = -1.) Determine whether the linear transformation T is invertible.

Totally lost on this question :/ Please show some example matrix and how it is transformed as the question asks if possible. I want to see what actually happens to the elements in the matrix rather than the answer (think that would be more important)

A(Matrix Elements):
This is my interpretation:

A = <math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}</math> where <math>a_{ij} \in C</math>, then B = <math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\ia_{21}+a_{31}&ia_{22}+a_{32}\end{pmatrix}</math>.

Therefore, T(A) = Bt is T<math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}</math>=<math>\begin{pmatrix}a_{11}&a_{21}&ia_{21}+a_{31}\\a_{12}&a_{22}&ia_{22}+a_{32}\end{pmatrix}</math>

R: Thx alot, the matricies are really helpful :)

===Sec. 2.4 Lemma p. 101===
Q: In the proof of the lemma, the second line, we have "T(beta) spans R(T) = W". How do we know that R(T) = W? This would be true if dim V = dim W, because then T would be onto, but we can't assume what we're trying to prove.

A: The first line of the Lemma states, "Let T be an '''invertible''' linear trans..." So, T is onto(and 1-1), thus "T(beta) spans R(T) = W".

R: Yes. My trouble was with the fact that invertibility implies onto-ness. I thought that if we had <math> T:P_2 (R)->P_6(R) </math>, and T(f) = xf, then T would still be invertible since you can 'recover' the f if you were given xf. I guess it makes more sense to not call T invertible in this case, because <math>T^{-1}</math> is technically only one-to-one over the range of T.

R: T has to be both Onto and 1-1 so that it's invertible. In your example, some of the <math> f \in P_6(R)</math> will not be 'recovered' because they weren't mapped from <math> P_2(R)</math>. Furthermore, T-1 has to map the whole vector space W back to V(as defined on p.99) but not the range only. In other words, if T is 1-1 only, <math>T^{-1} \circ T(v) = v, \forall v\in V</math> but <math>T \circ T^{-1}(w) \neq w, \exists w\in W</math>, because some T-1(w) are not defined.

R: That nicely rigorizes what I was thinking, and I'm convinced. Thanks.

===Exam April/May 2006 #7===
Q: Let T : V -> V and U : V -> V be linear operators on a finite-dimensional vector space V. Assume that U is invertible and T is diagonalizable. Prove that the linear operator UTU-1 = U o T o U-1.

I dont know where or how to start this question ><.

A: I think we need to prove that UTU-1 is diagonalizable instead of proving UTU-1 = U o T o U-1.

I started by letting A = UTU-1, then multiplying U-1 and U to the both sides, we get U-1AU = U-1UTU-1U iff U-1AU = T. Since T is diagonalizable, therefore there exists an invertible matrix Q s.t Q-1TQ = D, where D is a diagonal matrix. Therefore, Q-1(U-1AU)Q = D iff (UQ)-1A(UQ) = D (because U and Q invertible, Q-1U-1 = (UQ)-1), it follows that A is diagonalizable.

06-240/Final Exam Preparation Forum

2006-12-13T00:24:10Z

Wanmike: /* Exam April 2004 #6(a) */

{{06-240/Navigation}}

If you have questions, ask them here and hopefully someone else will know the answer. (Answering questions will probably help you understand it more).

Since many of us (including I) don't really know how to use Wiki's, I suggest that we keep the formatting simple: I will post a template at the top of this page, and if you want to add something just click on the "edit", copy the template, and insert your question. Order the questions according to section (i.e. solved/unsolved; '''whoever created the question must decide if it is solved'''), with the newest at the top, except for the template question. In general, I wouldn't retype the question if it's from the book because that's tedious and we all have the book.

(By the way, I think you leave a space between lines in the code to make a new line; that is, simply pressing enter once will not make a new line. Also, you can press a button at the top of the editing textbox that lets you put in simple equations.)

==Unsolved Questions==

===Question Template===
Q: Can someone help me prove: "If an [[integer]] <math>n</math> is greater than 2, then <math>a^n + b^n = c^n</math> has no solutions in non-zero integers <math>a</math>, <math>b</math>, and <math>c</math>."? I had the answer in my head at one point, but the margins of the piece of paper I was working with was too small to fit it.

===Rank of Matrices===
Q: Prove that if rankA = 0 (for A with dimension m x n), then A is the zero matrix.
(Question is found on p 166 (#3))
Did anyone use transformations in this?
My proof relies on rank(L_A) = 0 (left-multiplication transformation)implying L_A is the zero transformation.
Is there an easier way?

A: Depends on what you're allowed to assume. If you can use that the rank is equal to the number of linearly independent columns, then clearly all of the columns must be zero (otherwise you have at least one linearly independent vector).

===Determinants===
Q : If we can make same matrix with 2n-1 times of row swaps, what does it mean ? Does it mean that determinant is 0 ?

R: Is this related to a question somewhere?

===Sec 3.2 Ex. 19===
Q: "Let A be an m x n matrix with rank m and B be an n x p matrix with rank n. Determine the rank of AB. Justify your answer." I know how to find that the rank can't be more than m (not much of an accomplishment), but I can't finish it.

A: According to Theorem 3.7(a),(c)&(d)(p.159), I would say rank(AB) <math>\le </math>min(m, n).

R: Can we not get any more specific than that?

===Sec. 3.2 Ex. 21===
Q: "Let A be an m x n matrix with rank m. Prove that there exists an n x m matrix B such that AB=<math>I_m</math>".

===Exam April/May 2006 #4===
Q: Suppose that A, B Є Mmxn(F), and rank(A) = rank (B). Prove that there exist invertible matrices P Є Mmxm(F) and Q Є Mnxn(F) such that B = PAQ.

A(partial): Here is a sketch. If you rref A and B by applying a series of elementary row operation matricies, they will both look similar. That is, they will have a section of 1's and 0's (each 1 is the only number in its column) and then a section of "remaining stuff", and these sections will be the same "size" because their ranks are the same. Then, using the elementary column matrix operations, you can essentially modify the "remaining stuff" as much as you like, by adding multiples of the "nice" columns (with single-1's). These row and column operations can then be grouped nicely and set to be equal to P and Q, which are invertible because products of elementary matricies are invertible.

I know this is very rough, but even if I did have a full answer I wouldn't now how to typeset it.

===Complex Numbers===
Q: If 'C' is used in the context of a vector space (as in "define T:C->C"), then should we consider C to be the vector space of C over the field C, or instead C over the field R?

===Readings?===
Q: Are we expected to section 5.2 of the textbook? Although the Assignments tell us to read it, we didn't do any questions, or cover it in class.

A: I highly doubt it; hopefully someone will ask Prof. Bar-Natan tomorrow and post the answer here. There were a few other chapters that had sections we never really talked about either (some applications). Addendum: I second this request for a slight narrowing of what the relevant readings are--for instance, can we be more efficient in our reading of chapter 4 somehow?

R: I think that if you want to cut down on Chapter 4, then skipping applications of area (discussed very briefly in class) and determinants of order 2 is the most you can do.

R: What about 4.5, the axiomatic details. It discusses how the determinant is uniquely defined by the three axiomatic properties, but I don't think we did that in class.

===Exam April 2004 #6(a)===
Q: Suppose A is an invertible matrix for which the sum of entries of each row is a scalar <math>\lambda</math>. Show that the sum of entries of each row of A-1 is <math>1/\lambda</math>. (Hint: find an eigenvector for A with eigenvalue <math>\lambda</math>.)
If A is a diagonal matrix, then it's obvious that the sum of entries of each row is <math>\lambda</math> and the sum of entries of each row of A-1 is <math>1/\lambda</math>. I was stuck with a more general invertible matrix.

A: Following the hint, you can see that an eigenvector corresponding to <math>\lambda</math> is (1, 1, 1, ...)*. Therefore <math> Av=\lambda v</math>, and rearranging you get <math> A^{-1}v=1/\lambda v</math>. Using the same logic as before, you can show that since this <math>\lambda</math> corresponds to a homogeneous system of equations with the same eigenvector v = (1, 1, 1, ...), the sum of each row is equal to <math>1/\lambda</math>.

*Just to elaborate on the first part, you are looking for a vector <math> v = (x_1, x_2, x_3, ...) </math> so that <math> A-\lambda I = 0</math>. This corresponds to the system:
<math>\begin{pmatrix}(a_{11}-\lambda)x_1&a_{12}x_2&a_{13}x_3&...\\a_{21}x_1&(a_{22}-\lambda)x_2&a_{23}x_3&...\\
a_{11}x_1&a_{12}x_1&(a_{13}-\lambda)x_3&...\end{pmatrix}</math>,
and so in each row you can see that <math>x_1=1, x_2=1, x_3=3</math> works because then all the a's in each row add up to <math>\lambda</math>.

==Solved Questions==

===Question Template===
Q: How many ways are there to get to the nth stair, if at each step you can move either one or two squares up?

A: This question can be easily modeled by the Fibonacci numbers, with the nth number being the ways to get to the nth stair. This is because, to get to the nth stair, you can come only from the n-1th or the n-2th. This is exactly how the Fibonacci numbers are defined; the proof is simple by induction.

===Sec. 1.3 Thm 1.3 Proof===
Q: In the first paragraph of the proof, it says "But also x + 0 = x , and thus 0'=0." How do we know 0 (that is 0 of V) even exists in W? I understand that we know ''some'' zero exists (0'), but not why ''the'' zero (0) exists.

A: x is in W as well in V. Thus, x + 0 = x (VS 3).

Reply: Oh I see... now it looks so obvious =/. Thanks.

===Exam April/May 2006 #3(b)===
Q: Let T : M3x2(C) -> M2x3(C) be defined as follows. Given A Є M3x2(C), let B be the matrix obtained from A by adding i times the second row of A to the third row of A. Let T(A) = Bt, where Bt is the transpose of B. (Note: Here, i is a complex number such that i2 = -1.) Determine whether the linear transformation T is invertible.

Totally lost on this question :/ Please show some example matrix and how it is transformed as the question asks if possible. I want to see what actually happens to the elements in the matrix rather than the answer (think that would be more important)

A(Matrix Elements):
This is my interpretation:

A = <math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}</math> where <math>a_{ij} \in C</math>, then B = <math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\ia_{21}+a_{31}&ia_{22}+a_{32}\end{pmatrix}</math>.

Therefore, T(A) = Bt is T<math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}</math>=<math>\begin{pmatrix}a_{11}&a_{21}&ia_{21}+a_{31}\\a_{12}&a_{22}&ia_{22}+a_{32}\end{pmatrix}</math>

R: Thx alot, the matricies are really helpful :)

===Sec. 2.4 Lemma p. 101===
Q: In the proof of the lemma, the second line, we have "T(beta) spans R(T) = W". How do we know that R(T) = W? This would be true if dim V = dim W, because then T would be onto, but we can't assume what we're trying to prove.

A: The first line of the Lemma states, "Let T be an '''invertible''' linear trans..." So, T is onto(and 1-1), thus "T(beta) spans R(T) = W".

R: Yes. My trouble was with the fact that invertibility implies onto-ness. I thought that if we had <math> T:P_2 (R)->P_6(R) </math>, and T(f) = xf, then T would still be invertible since you can 'recover' the f if you were given xf. I guess it makes more sense to not call T invertible in this case, because <math>T^{-1}</math> is technically only one-to-one over the range of T.

R: T has to be both Onto and 1-1 so that it's invertible. In your example, some of the <math> f \in P_6(R)</math> will not be 'recovered' because they weren't mapped from <math> P_2(R)</math>. Furthermore, T-1 has to map the whole vector space W back to V(as defined on p.99) but not the range only. In other words, if T is 1-1 only, <math>T^{-1} \circ T(v) = v, \forall v\in V</math> but <math>T \circ T^{-1}(w) \neq w, \exists w\in W</math>, because some T-1(w) are not defined.

R: That nicely rigorizes what I was thinking, and I'm convinced. Thanks.

===Exam April/May 2006 #7===
Q: Let T : V -> V and U : V -> V be linear operators on a finite-dimensional vector space V. Assume that U is invertible and T is diagonalizable. Prove that the linear operator UTU-1 = U o T o U-1.

I dont know where or how to start this question ><.

A: I think we need to prove that UTU-1 is diagonalizable instead of proving UTU-1 = U o T o U-1.

I started by letting A = UTU-1, then multiplying U-1 and U to the both sides, we get U-1AU = U-1UTU-1U iff U-1AU = T. Since T is diagonalizable, therefore there exists an invertible matrix Q s.t Q-1TQ = D, where D is a diagonal matrix. Therefore, Q-1(U-1AU)Q = D iff (UQ)-1A(UQ) = D (because U and Q invertible, Q-1U-1 = (UQ)-1), it follows that A is diagonalizable.

06-240/Final Exam Preparation Forum

2006-12-13T00:11:20Z

Wanmike: /* Exam April 2004 #6(a) */

{{06-240/Navigation}}

If you have questions, ask them here and hopefully someone else will know the answer. (Answering questions will probably help you understand it more).

Since many of us (including I) don't really know how to use Wiki's, I suggest that we keep the formatting simple: I will post a template at the top of this page, and if you want to add something just click on the "edit", copy the template, and insert your question. Order the questions according to section (i.e. solved/unsolved; '''whoever created the question must decide if it is solved'''), with the newest at the top, except for the template question. In general, I wouldn't retype the question if it's from the book because that's tedious and we all have the book.

(By the way, I think you leave a space between lines in the code to make a new line; that is, simply pressing enter once will not make a new line. Also, you can press a button at the top of the editing textbox that lets you put in simple equations.)

==Unsolved Questions==

===Question Template===
Q: Can someone help me prove: "If an [[integer]] <math>n</math> is greater than 2, then <math>a^n + b^n = c^n</math> has no solutions in non-zero integers <math>a</math>, <math>b</math>, and <math>c</math>."? I had the answer in my head at one point, but the margins of the piece of paper I was working with was too small to fit it.

===Rank of Matrices===
Q: Prove that if rankA = 0 (for A with dimension m x n), then A is the zero matrix.
(Question is found on p 166 (#3))
Did anyone use transformations in this?
My proof relies on rank(L_A) = 0 (left-multiplication transformation)implying L_A is the zero transformation.
Is there an easier way?

A: Depends on what you're allowed to assume. If you can use that the rank is equal to the number of linearly independent columns, then clearly all of the columns must be zero (otherwise you have at least one linearly independent vector).

===Determinants===
Q : If we can make same matrix with 2n-1 times of row swaps, what does it mean ? Does it mean that determinant is 0 ?

R: Is this related to a question somewhere?

===Sec 3.2 Ex. 19===
Q: "Let A be an m x n matrix with rank m and B be an n x p matrix with rank n. Determine the rank of AB. Justify your answer." I know how to find that the rank can't be more than m (not much of an accomplishment), but I can't finish it.

A: According to Theorem 3.7(a),(c)&(d)(p.159), I would say rank(AB) <math>\le </math>min(m, n).

R: Can we not get any more specific than that?

===Sec. 3.2 Ex. 21===
Q: "Let A be an m x n matrix with rank m. Prove that there exists an n x m matrix B such that AB=<math>I_m</math>".

===Exam April/May 2006 #4===
Q: Suppose that A, B Є Mmxn(F), and rank(A) = rank (B). Prove that there exist invertible matrices P Є Mmxm(F) and Q Є Mnxn(F) such that B = PAQ.

A(partial): Here is a sketch. If you rref A and B by applying a series of elementary row operation matricies, they will both look similar. That is, they will have a section of 1's and 0's (each 1 is the only number in its column) and then a section of "remaining stuff", and these sections will be the same "size" because their ranks are the same. Then, using the elementary column matrix operations, you can essentially modify the "remaining stuff" as much as you like, by adding multiples of the "nice" columns (with single-1's). These row and column operations can then be grouped nicely and set to be equal to P and Q, which are invertible because products of elementary matricies are invertible.

I know this is very rough, but even if I did have a full answer I wouldn't now how to typeset it.

===Complex Numbers===
Q: If 'C' is used in the context of a vector space (as in "define T:C->C"), then should we consider C to be the vector space of C over the field C, or instead C over the field R?

===Readings?===
Q: Are we expected to section 5.2 of the textbook? Although the Assignments tell us to read it, we didn't do any questions, or cover it in class.

A: I highly doubt it; hopefully someone will ask Prof. Bar-Natan tomorrow and post the answer here. There were a few other chapters that had sections we never really talked about either (some applications). Addendum: I second this request for a slight narrowing of what the relevant readings are--for instance, can we be more efficient in our reading of chapter 4 somehow?

R: I think that if you want to cut down on Chapter 4, then skipping applications of area (discussed very briefly in class) and determinants of order 2 is the most you can do.

R: What about 4.5, the axiomatic details. It discusses how the determinant is uniquely defined by the three axiomatic properties, but I don't think we did that in class.

===Exam April 2004 #6(a)===
Q: Suppose A is an invertible matrix for which the sum of entries of each row is a scalar <math>\lambda</math>. Show that the sum of entries of each row of A-1 is <math>1/\lambda</math>. (Hint: find an eigenvector for A with eigenvalue <math>\lambda</math>.)
If A is a diagonal matrix, then it's obvious that the sum of entries of each row is <math>\lambda</math> and the sum of entries of each row of A-1 is <math>1/\lambda</math>. I was stuck with a more general invertible matrix.

A: Following the hint, you can see that an eigenvector corresponding to <math>\lambda</math> is (1, 1, 1, ...). Therefore <math> Av=\lambda v</math>, and rearranging you get <math> A^{-1}v=1/\lambda v</math>. Using the same logic as before, you can show that since this <math>\lambda</math> corresponds to a homogeneous system of equations with the same eigenvector v = (1, 1, 1, ...), the sum of each row is equal to <math>1/\lambda</math>.

==Solved Questions==

===Question Template===
Q: How many ways are there to get to the nth stair, if at each step you can move either one or two squares up?

A: This question can be easily modeled by the Fibonacci numbers, with the nth number being the ways to get to the nth stair. This is because, to get to the nth stair, you can come only from the n-1th or the n-2th. This is exactly how the Fibonacci numbers are defined; the proof is simple by induction.

===Sec. 1.3 Thm 1.3 Proof===
Q: In the first paragraph of the proof, it says "But also x + 0 = x , and thus 0'=0." How do we know 0 (that is 0 of V) even exists in W? I understand that we know ''some'' zero exists (0'), but not why ''the'' zero (0) exists.

A: x is in W as well in V. Thus, x + 0 = x (VS 3).

Reply: Oh I see... now it looks so obvious =/. Thanks.

===Exam April/May 2006 #3(b)===
Q: Let T : M3x2(C) -> M2x3(C) be defined as follows. Given A Є M3x2(C), let B be the matrix obtained from A by adding i times the second row of A to the third row of A. Let T(A) = Bt, where Bt is the transpose of B. (Note: Here, i is a complex number such that i2 = -1.) Determine whether the linear transformation T is invertible.

Totally lost on this question :/ Please show some example matrix and how it is transformed as the question asks if possible. I want to see what actually happens to the elements in the matrix rather than the answer (think that would be more important)

A(Matrix Elements):
This is my interpretation:

A = <math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}</math> where <math>a_{ij} \in C</math>, then B = <math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\ia_{21}+a_{31}&ia_{22}+a_{32}\end{pmatrix}</math>.

Therefore, T(A) = Bt is T<math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}</math>=<math>\begin{pmatrix}a_{11}&a_{21}&ia_{21}+a_{31}\\a_{12}&a_{22}&ia_{22}+a_{32}\end{pmatrix}</math>

R: Thx alot, the matricies are really helpful :)

===Sec. 2.4 Lemma p. 101===
Q: In the proof of the lemma, the second line, we have "T(beta) spans R(T) = W". How do we know that R(T) = W? This would be true if dim V = dim W, because then T would be onto, but we can't assume what we're trying to prove.

A: The first line of the Lemma states, "Let T be an '''invertible''' linear trans..." So, T is onto(and 1-1), thus "T(beta) spans R(T) = W".

R: Yes. My trouble was with the fact that invertibility implies onto-ness. I thought that if we had <math> T:P_2 (R)->P_6(R) </math>, and T(f) = xf, then T would still be invertible since you can 'recover' the f if you were given xf. I guess it makes more sense to not call T invertible in this case, because <math>T^{-1}</math> is technically only one-to-one over the range of T.

R: T has to be both Onto and 1-1 so that it's invertible. In your example, some of the <math> f \in P_6(R)</math> will not be 'recovered' because they weren't mapped from <math> P_2(R)</math>. Furthermore, T-1 has to map the whole vector space W back to V(as defined on p.99) but not the range only. In other words, if T is 1-1 only, <math>T^{-1} \circ T(v) = v, \forall v\in V</math> but <math>T \circ T^{-1}(w) \neq w, \exists w\in W</math>, because some T-1(w) are not defined.

R: That nicely rigorizes what I was thinking, and I'm convinced. Thanks.

===Exam April/May 2006 #7===
Q: Let T : V -> V and U : V -> V be linear operators on a finite-dimensional vector space V. Assume that U is invertible and T is diagonalizable. Prove that the linear operator UTU-1 = U o T o U-1.

I dont know where or how to start this question ><.

A: I think we need to prove that UTU-1 is diagonalizable instead of proving UTU-1 = U o T o U-1.

I started by letting A = UTU-1, then multiplying U-1 and U to the both sides, we get U-1AU = U-1UTU-1U iff U-1AU = T. Since T is diagonalizable, therefore there exists an invertible matrix Q s.t Q-1TQ = D, where D is a diagonal matrix. Therefore, Q-1(U-1AU)Q = D iff (UQ)-1A(UQ) = D (because U and Q invertible, Q-1U-1 = (UQ)-1), it follows that A is diagonalizable.

06-240/Final Exam Preparation Forum

2006-12-13T00:05:30Z

Wanmike: /* Exam April 2004 #6(a) */

{{06-240/Navigation}}

If you have questions, ask them here and hopefully someone else will know the answer. (Answering questions will probably help you understand it more).

Since many of us (including I) don't really know how to use Wiki's, I suggest that we keep the formatting simple: I will post a template at the top of this page, and if you want to add something just click on the "edit", copy the template, and insert your question. Order the questions according to section (i.e. solved/unsolved; '''whoever created the question must decide if it is solved'''), with the newest at the top, except for the template question. In general, I wouldn't retype the question if it's from the book because that's tedious and we all have the book.

(By the way, I think you leave a space between lines in the code to make a new line; that is, simply pressing enter once will not make a new line. Also, you can press a button at the top of the editing textbox that lets you put in simple equations.)

==Unsolved Questions==

===Question Template===
Q: Can someone help me prove: "If an [[integer]] <math>n</math> is greater than 2, then <math>a^n + b^n = c^n</math> has no solutions in non-zero integers <math>a</math>, <math>b</math>, and <math>c</math>."? I had the answer in my head at one point, but the margins of the piece of paper I was working with was too small to fit it.

===Rank of Matrices===
Q: Prove that if rankA = 0 (for A with dimension m x n), then A is the zero matrix.
(Question is found on p 166 (#3))
Did anyone use transformations in this?
My proof relies on rank(L_A) = 0 (left-multiplication transformation)implying L_A is the zero transformation.
Is there an easier way?

A: Depends on what you're allowed to assume. If you can use that the rank is equal to the number of linearly independent columns, then clearly all of the columns must be zero (otherwise you have at least one linearly independent vector).

===Determinants===
Q : If we can make same matrix with 2n-1 times of row swaps, what does it mean ? Does it mean that determinant is 0 ?

R: Is this related to a question somewhere?

===Sec 3.2 Ex. 19===
Q: "Let A be an m x n matrix with rank m and B be an n x p matrix with rank n. Determine the rank of AB. Justify your answer." I know how to find that the rank can't be more than m (not much of an accomplishment), but I can't finish it.

A: According to Theorem 3.7(a),(c)&(d)(p.159), I would say rank(AB) <math>\le </math>min(m, n).

R: Can we not get any more specific than that?

===Sec. 3.2 Ex. 21===
Q: "Let A be an m x n matrix with rank m. Prove that there exists an n x m matrix B such that AB=<math>I_m</math>".

===Exam April/May 2006 #4===
Q: Suppose that A, B Є Mmxn(F), and rank(A) = rank (B). Prove that there exist invertible matrices P Є Mmxm(F) and Q Є Mnxn(F) such that B = PAQ.

A(partial): Here is a sketch. If you rref A and B by applying a series of elementary row operation matricies, they will both look similar. That is, they will have a section of 1's and 0's (each 1 is the only number in its column) and then a section of "remaining stuff", and these sections will be the same "size" because their ranks are the same. Then, using the elementary column matrix operations, you can essentially modify the "remaining stuff" as much as you like, by adding multiples of the "nice" columns (with single-1's). These row and column operations can then be grouped nicely and set to be equal to P and Q, which are invertible because products of elementary matricies are invertible.

I know this is very rough, but even if I did have a full answer I wouldn't now how to typeset it.

===Complex Numbers===
Q: If 'C' is used in the context of a vector space (as in "define T:C->C"), then should we consider C to be the vector space of C over the field C, or instead C over the field R?

===Readings?===
Q: Are we expected to section 5.2 of the textbook? Although the Assignments tell us to read it, we didn't do any questions, or cover it in class.

A: I highly doubt it; hopefully someone will ask Prof. Bar-Natan tomorrow and post the answer here. There were a few other chapters that had sections we never really talked about either (some applications). Addendum: I second this request for a slight narrowing of what the relevant readings are--for instance, can we be more efficient in our reading of chapter 4 somehow?

R: I think that if you want to cut down on Chapter 4, then skipping applications of area (discussed very briefly in class) and determinants of order 2 is the most you can do.

R: What about 4.5, the axiomatic details. It discusses how the determinant is uniquely defined by the three axiomatic properties, but I don't think we did that in class.

===Exam April 2004 #6(a)===
Q: Suppose A is an invertible matrix for which the sum of entries of each row is a scalar <math>\lambda</math>. Show that the sum of entries of each row of A-1 is <math>1/\lambda</math>. (Hint: find an eigenvector for A with eigenvalue <math>\lambda</math>.)
If A is a diagonal matrix, then it's obvious that the sum of entries of each row is <math>\lambda</math> and the sum of entries of each row of A-1 is <math>1/\lambda</math>. I was stuck with a more general invertible matrix.

A: Looks tough. I'll work on it, but for now I can say that, to answer the hint, the vector (1, 1, 1, 1, ...) is an eigenvector.

==Solved Questions==

===Question Template===
Q: How many ways are there to get to the nth stair, if at each step you can move either one or two squares up?

A: This question can be easily modeled by the Fibonacci numbers, with the nth number being the ways to get to the nth stair. This is because, to get to the nth stair, you can come only from the n-1th or the n-2th. This is exactly how the Fibonacci numbers are defined; the proof is simple by induction.

===Sec. 1.3 Thm 1.3 Proof===
Q: In the first paragraph of the proof, it says "But also x + 0 = x , and thus 0'=0." How do we know 0 (that is 0 of V) even exists in W? I understand that we know ''some'' zero exists (0'), but not why ''the'' zero (0) exists.

A: x is in W as well in V. Thus, x + 0 = x (VS 3).

Reply: Oh I see... now it looks so obvious =/. Thanks.

===Exam April/May 2006 #3(b)===
Q: Let T : M3x2(C) -> M2x3(C) be defined as follows. Given A Є M3x2(C), let B be the matrix obtained from A by adding i times the second row of A to the third row of A. Let T(A) = Bt, where Bt is the transpose of B. (Note: Here, i is a complex number such that i2 = -1.) Determine whether the linear transformation T is invertible.

Totally lost on this question :/ Please show some example matrix and how it is transformed as the question asks if possible. I want to see what actually happens to the elements in the matrix rather than the answer (think that would be more important)

A(Matrix Elements):
This is my interpretation:

A = <math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}</math> where <math>a_{ij} \in C</math>, then B = <math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\ia_{21}+a_{31}&ia_{22}+a_{32}\end{pmatrix}</math>.

Therefore, T(A) = Bt is T<math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}</math>=<math>\begin{pmatrix}a_{11}&a_{21}&ia_{21}+a_{31}\\a_{12}&a_{22}&ia_{22}+a_{32}\end{pmatrix}</math>

R: Thx alot, the matricies are really helpful :)

===Sec. 2.4 Lemma p. 101===
Q: In the proof of the lemma, the second line, we have "T(beta) spans R(T) = W". How do we know that R(T) = W? This would be true if dim V = dim W, because then T would be onto, but we can't assume what we're trying to prove.

A: The first line of the Lemma states, "Let T be an '''invertible''' linear trans..." So, T is onto(and 1-1), thus "T(beta) spans R(T) = W".

R: Yes. My trouble was with the fact that invertibility implies onto-ness. I thought that if we had <math> T:P_2 (R)->P_6(R) </math>, and T(f) = xf, then T would still be invertible since you can 'recover' the f if you were given xf. I guess it makes more sense to not call T invertible in this case, because <math>T^{-1}</math> is technically only one-to-one over the range of T.

R: T has to be both Onto and 1-1 so that it's invertible. In your example, some of the <math> f \in P_6(R)</math> will not be 'recovered' because they weren't mapped from <math> P_2(R)</math>. Furthermore, T-1 has to map the whole vector space W back to V(as defined on p.99) but not the range only. In other words, if T is 1-1 only, <math>T^{-1} \circ T(v) = v, \forall v\in V</math> but <math>T \circ T^{-1}(w) \neq w, \exists w\in W</math>, because some T-1(w) are not defined.

R: That nicely rigorizes what I was thinking, and I'm convinced. Thanks.

===Exam April/May 2006 #7===
Q: Let T : V -> V and U : V -> V be linear operators on a finite-dimensional vector space V. Assume that U is invertible and T is diagonalizable. Prove that the linear operator UTU-1 = U o T o U-1.

I dont know where or how to start this question ><.

A: I think we need to prove that UTU-1 is diagonalizable instead of proving UTU-1 = U o T o U-1.

I started by letting A = UTU-1, then multiplying U-1 and U to the both sides, we get U-1AU = U-1UTU-1U iff U-1AU = T. Since T is diagonalizable, therefore there exists an invertible matrix Q s.t Q-1TQ = D, where D is a diagonal matrix. Therefore, Q-1(U-1AU)Q = D iff (UQ)-1A(UQ) = D (because U and Q invertible, Q-1U-1 = (UQ)-1), it follows that A is diagonalizable.

06-240/Final Exam Preparation Forum

2006-12-12T23:45:28Z

Wanmike: /* Sec. 2.4 Lemma p. 101 */

{{06-240/Navigation}}

If you have questions, ask them here and hopefully someone else will know the answer. (Answering questions will probably help you understand it more).

Since many of us (including I) don't really know how to use Wiki's, I suggest that we keep the formatting simple: I will post a template at the top of this page, and if you want to add something just click on the "edit", copy the template, and insert your question. Order the questions according to section (i.e. solved/unsolved; '''whoever created the question must decide if it is solved'''), with the newest at the top, except for the template question. In general, I wouldn't retype the question if it's from the book because that's tedious and we all have the book.

(By the way, I think you leave a space between lines in the code to make a new line; that is, simply pressing enter once will not make a new line. Also, you can press a button at the top of the editing textbox that lets you put in simple equations.)

==Unsolved Questions==

===Question Template===
Q: Can someone help me prove: "If an [[integer]] <math>n</math> is greater than 2, then <math>a^n + b^n = c^n</math> has no solutions in non-zero integers <math>a</math>, <math>b</math>, and <math>c</math>."? I had the answer in my head at one point, but the margins of the piece of paper I was working with was too small to fit it.

===Rank of Matrices===
Q: Prove that if rankA = 0 (for A with dimension m x n), then A is the zero matrix.
(Question is found on p 166 (#3))
Did anyone use transformations in this?
My proof relies on rank(L_A) = 0 (left-multiplication transformation)implying L_A is the zero transformation.
Is there an easier way?

A: Depends on what you're allowed to assume. If you can use that the rank is equal to the number of linearly independent columns, then clearly all of the columns must be zero (otherwise you have at least one linearly independent vector).

===Determinants===
Q : If we can make same matrix with 2n-1 times of row swaps, what does it mean ? Does it mean that determinant is 0 ?

R: Is this related to a question somewhere?

===Sec 3.2 Ex. 19===
Q: "Let A be an m x n matrix with rank m and B be an n x p matrix with rank n. Determine the rank of AB. Justify your answer." I know how to find that the rank can't be more than m (not much of an accomplishment), but I can't finish it.

A: According to Theorem 3.7(a),(c)&(d)(p.159), I would say rank(AB) <math>\le </math>min(m, n).

R: Can we not get any more specific than that?

===Sec. 3.2 Ex. 21===
Q: "Let A be an m x n matrix with rank m. Prove that there exists an n x m matrix B such that AB=<math>I_m</math>".

===Exam April/May 2006 #4===
Q: Suppose that A, B Є Mmxn(F), and rank(A) = rank (B). Prove that there exist invertible matrices P Є Mmxm(F) and Q Є Mnxn(F) such that B = PAQ.

A(partial): Here is a sketch. If you rref A and B by applying a series of elementary row operation matricies, they will both look similar. That is, they will have a section of 1's and 0's (each 1 is the only number in its column) and then a section of "remaining stuff", and these sections will be the same "size" because their ranks are the same. Then, using the elementary column matrix operations, you can essentially modify the "remaining stuff" as much as you like, by adding multiples of the "nice" columns (with single-1's). These row and column operations can then be grouped nicely and set to be equal to P and Q, which are invertible because products of elementary matricies are invertible.

I know this is very rough, but even if I did have a full answer I wouldn't now how to typeset it.

===Complex Numbers===
Q: If 'C' is used in the context of a vector space (as in "define T:C->C"), then should we consider C to be the vector space of C over the field C, or instead C over the field R?

===Readings?===
Q: Are we expected to section 5.2 of the textbook? Although the Assignments tell us to read it, we didn't do any questions, or cover it in class.

A: I highly doubt it; hopefully someone will ask Prof. Bar-Natan tomorrow and post the answer here. There were a few other chapters that had sections we never really talked about either (some applications). Addendum: I second this request for a slight narrowing of what the relevant readings are--for instance, can we be more efficient in our reading of chapter 4 somehow?

R: I think that if you want to cut down on Chapter 4, then skipping applications of area (discussed very briefly in class) and determinants of order 2 is the most you can do.

R: What about 4.5, the axiomatic details. It discusses how the determinant is uniquely defined by the three axiomatic properties, but I don't think we did that in class.

===Exam April 2004 #6(a)===
Q: Suppose A is an invertible matrix for which the sum of entries of each row is a scalar <math>\lambda</math>. Show that the sum of entries of each row of A-1 is <math>1/\lambda</math>. (Hint: find an eigenvector for A with eigenvalue <math>\lambda</math>.)
If A is a diagonal matrix, then it's obvious that the sum of entries of each row is <math>\lambda</math> and the sum of entries of each row of A-1 is <math>1/\lambda</math>. I was stuck with a more general invertible matrix.

==Solved Questions==

===Question Template===
Q: How many ways are there to get to the nth stair, if at each step you can move either one or two squares up?

A: This question can be easily modeled by the Fibonacci numbers, with the nth number being the ways to get to the nth stair. This is because, to get to the nth stair, you can come only from the n-1th or the n-2th. This is exactly how the Fibonacci numbers are defined; the proof is simple by induction.

===Sec. 1.3 Thm 1.3 Proof===
Q: In the first paragraph of the proof, it says "But also x + 0 = x , and thus 0'=0." How do we know 0 (that is 0 of V) even exists in W? I understand that we know ''some'' zero exists (0'), but not why ''the'' zero (0) exists.

A: x is in W as well in V. Thus, x + 0 = x (VS 3).

Reply: Oh I see... now it looks so obvious =/. Thanks.

===Exam April/May 2006 #3(b)===
Q: Let T : M3x2(C) -> M2x3(C) be defined as follows. Given A Є M3x2(C), let B be the matrix obtained from A by adding i times the second row of A to the third row of A. Let T(A) = Bt, where Bt is the transpose of B. (Note: Here, i is a complex number such that i2 = -1.) Determine whether the linear transformation T is invertible.

Totally lost on this question :/ Please show some example matrix and how it is transformed as the question asks if possible. I want to see what actually happens to the elements in the matrix rather than the answer (think that would be more important)

A(Matrix Elements):
This is my interpretation:

A = <math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}</math> where <math>a_{ij} \in C</math>, then B = <math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\ia_{21}+a_{31}&ia_{22}+a_{32}\end{pmatrix}</math>.

Therefore, T(A) = Bt is T<math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}</math>=<math>\begin{pmatrix}a_{11}&a_{21}&ia_{21}+a_{31}\\a_{12}&a_{22}&ia_{22}+a_{32}\end{pmatrix}</math>

R: Thx alot, the matricies are really helpful :)

===Sec. 2.4 Lemma p. 101===
Q: In the proof of the lemma, the second line, we have "T(beta) spans R(T) = W". How do we know that R(T) = W? This would be true if dim V = dim W, because then T would be onto, but we can't assume what we're trying to prove.

A: The first line of the Lemma states, "Let T be an '''invertible''' linear trans..." So, T is onto(and 1-1), thus "T(beta) spans R(T) = W".

R: Yes. My trouble was with the fact that invertibility implies onto-ness. I thought that if we had <math> T:P_2 (R)->P_6(R) </math>, and T(f) = xf, then T would still be invertible since you can 'recover' the f if you were given xf. I guess it makes more sense to not call T invertible in this case, because <math>T^{-1}</math> is technically only one-to-one over the range of T.

R: T has to be both Onto and 1-1 so that it's invertible. In your example, some of the <math> f \in P_6(R)</math> will not be 'recovered' because they weren't mapped from <math> P_2(R)</math>. Furthermore, T-1 has to map the whole vector space W back to V(as define on p.99) but not the range only. In other words, if T is 1-1 only, <math>T^{-1} \circ T(v) = v, \forall v\in V</math> but <math>T \circ T^{-1}(w) \neq w, \exists w\in W</math>, because some T-1(w) are undefined.

R: That nicely rigorizes what I was thinking, and I'm convinced. Thanks.

===Exam April/May 2006 #7===
Q: Let T : V -> V and U : V -> V be linear operators on a finite-dimensional vector space V. Assume that U is invertible and T is diagonalizable. Prove that the linear operator UTU-1 = U o T o U-1.

I dont know where or how to start this question ><.

A: I think we need to prove that UTU-1 is diagonalizable instead of proving UTU-1 = U o T o U-1.

I started by letting A = UTU-1, then multiplying U-1 and U to the both sides, we get U-1AU = U-1UTU-1U iff U-1AU = T. Since T is diagonalizable, therefore there exists an invertible matrix Q s.t Q-1TQ = D, where D is a diagonal matrix. Therefore, Q-1(U-1AU)Q = D iff (UQ)-1A(UQ) = D (because U and Q invertible, Q-1U-1 = (UQ)-1), it follows that A is diagonalizable.

06-240/Final Exam Preparation Forum

2006-12-12T20:33:58Z

Wanmike: /* Rank of Matrices */

{{06-240/Navigation}}

If you have questions, ask them here and hopefully someone else will know the answer. (Answering questions will probably help you understand it more).

Since many of us (including I) don't really know how to use Wiki's, I suggest that we keep the formatting simple: I will post a template at the top of this page, and if you want to add something just click on the "edit", copy the template, and insert your question. Order the questions according to section (i.e. solved/unsolved; '''whoever created the question must decide if it is solved'''), with the newest at the top, except for the template question. In general, I wouldn't retype the question if it's from the book because that's tedious and we all have the book.

(By the way, I think you leave a space between lines in the code to make a new line; that is, simply pressing enter once will not make a new line. Also, you can press a button at the top of the editing textbox that lets you put in simple equations.)

==Unsolved Questions==

===Question Template===
Q: Can someone help me prove: "If an [[integer]] <math>n</math> is greater than 2, then <math>a^n + b^n = c^n</math> has no solutions in non-zero integers <math>a</math>, <math>b</math>, and <math>c</math>."? I had the answer in my head at one point, but the margins of the piece of paper I was working with was too small to fit it.

===Rank of Matrices===
Q: Prove that if rankA = 0 (for A with dimension m x n), then A is the zero matrix.
(Question is found on p 166 (#3))
Did anyone use transformations in this?
My proof relies on rank(L_A) = 0 (left-multiplication transformation)implying L_A is the zero transformation.
Is there an easier way?

A: Depends on what you're allowed to assume. If you can use that the rank is equal to the number of linearly independent columns, then clearly all of the columns must be zero (otherwise you have at least one linearly independent vector).

===Determinants===
Q : If we can make same matrix with 2n-1 times of row swaps, what does it mean ? Does it mean that determinant is 0 ?

R: Is this related to a question somewhere?

===Sec 3.2 Ex. 19===
Q: "Let A be an m x n matrix with rank m and B be an n x p matrix with rank n. Determine the rank of AB. Justify your answer." I know how to find that the rank can't be more than m (not much of an accomplishment), but I can't finish it.

A: According to Theorem 3.7(a),(c)&(d)(p.159), I would say rank(AB) <math>\le </math>min(m, n).

R: Can we not get any more specific than that?

===Sec. 3.2 Ex. 21===
Q: "Let A be an m x n matrix with rank m. Prove that there exists an n x m matrix B such that AB=<math>I_m</math>".

===Exam April/May 2006 #4===
Q: Suppose that A, B Є Mmxn(F), and rank(A) = rank (B). Prove that there exist invertible matrices P Є Mmxm(F) and Q Є Mnxn(F) such that B = PAQ.

A(partial): Here is a sketch. If you rref A and B by applying a series of elementary row operation matricies, they will both look similar. That is, they will have a section of 1's and 0's (each 1 is the only number in its column) and then a section of "remaining stuff", and these sections will be the same "size" because their ranks are the same. Then, using the elementary column matrix operations, you can essentially modify the "remaining stuff" as much as you like, by adding multiples of the "nice" columns (with single-1's). These row and column operations can then be grouped nicely and set to be equal to P and Q, which are invertible because products of elementary matricies are invertible.

I know this is very rough, but even if I did have a full answer I wouldn't now how to typeset it.

===Complex Numbers===
Q: If 'C' is used in the context of a vector space (as in "define T:C->C"), then should we consider C to be the vector space of C over the field C, or instead C over the field R?

===Readings?===
Q: Are we expected to section 5.2 of the textbook? Although the Assignments tell us to read it, we didn't do any questions, or cover it in class.

A: I highly doubt it; hopefully someone will ask Prof. Bar-Natan tomorrow and post the answer here. There were a few other chapters that had sections we never really talked about either (some applications). Addendum: I second this request for a slight narrowing of what the relevant readings are--for instance, can we be more efficient in our reading of chapter 4 somehow?

R: I think that if you want to cut down on Chapter 4, then skipping applications of area (discussed very briefly in class) and determinants of order 2 is the most you can do.

R: What about 4.5, the axiomatic details. It discusses how the determinant is uniquely defined by the three axiomatic properties, but I don't think we did that in class.

==Solved Questions==

===Question Template===
Q: How many ways are there to get to the nth stair, if at each step you can move either one or two squares up?

A: This question can be easily modeled by the Fibonacci numbers, with the nth number being the ways to get to the nth stair. This is because, to get to the nth stair, you can come only from the n-1th or the n-2th. This is exactly how the Fibonacci numbers are defined; the proof is simple by induction.

===Sec. 1.3 Thm 1.3 Proof===
Q: In the first paragraph of the proof, it says "But also x + 0 = x , and thus 0'=0." How do we know 0 (that is 0 of V) even exists in W? I understand that we know ''some'' zero exists (0'), but not why ''the'' zero (0) exists.

A: x is in W as well in V. Thus, x + 0 = x (VS 3).

Reply: Oh I see... now it looks so obvious =/. Thanks.

===Exam April/May 2006 #3(b)===
Q: Let T : M3x2(C) -> M2x3(C) be defined as follows. Given A Є M3x2(C), let B be the matrix obtained from A by adding i times the second row of A to the third row of A. Let T(A) = Bt, where Bt is the transpose of B. (Note: Here, i is a complex number such that i2 = -1.) Determine whether the linear transformation T is invertible.

Totally lost on this question :/ Please show some example matrix and how it is transformed as the question asks if possible. I want to see what actually happens to the elements in the matrix rather than the answer (think that would be more important)

A(Matrix Elements):
This is my interpretation:

A = <math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}</math> where <math>a_{ij} \in C</math>, then B = <math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\ia_{21}+a_{31}&ia_{22}+a_{32}\end{pmatrix}</math>.

Therefore, T(A) = Bt is T<math>\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}</math>=<math>\begin{pmatrix}a_{11}&a_{21}&ia_{21}+a_{31}\\a_{12}&a_{22}&ia_{22}+a_{32}\end{pmatrix}</math>

R: Thx alot, the matricies are really helpful :)

===Sec. 2.4 Lemma p. 101===
Q: In the proof of the lemma, the second line, we have "T(beta) spans R(T) = W". How do we know that R(T) = W? This would be true if dim V = dim W, because then T would be onto, but we can't assume what we're trying to prove.

A: The first line of the Lemma states, "Let T be an '''invertible''' linear trans..." So, T is onto(and 1-1), thus "T(beta) spans R(T) = W".

R: Yes. My trouble was with the fact that imvertiblility implies onto-ness. I thought that if we had <math> T:P_2 (R)->P_6(R) </math>, and T(f) = xf, then T would still be invertible since you can 'recover' the f if you were given xf. I guess it makes more sense to not call T invertible in this case, because <math>T^{-1}</math> is technically only one-to-one over the range of T.

===Exam April/May 2006 #7===
Q: Let T : V -> V and U : V -> V be linear operators on a finite-dimensional vector space V. Assume that U is invertible and T is diagonalizable. Prove that the linear operator UTU-1 = U o T o U-1.

I dont know where or how to start this question ><.

A: I think we need to prove that UTU-1 is diagonalizable instead of proving UTU-1 = U o T o U-1.

I started by letting A = UTU-1, then multiplying U-1 and U to the both sides, we get U-1AU = U-1UTU-1U iff U-1AU = T. Since T is diagonalizable, therefore there exists an invertible matrix Q s.t Q-1TQ = D, where D is a diagonal matrix. Therefore, Q-1(U-1AU)Q = D iff (UQ)-1A(UQ) = D (because U and Q invertible, Q-1U-1 = (UQ)-1), it follows that A is diagonalizable.

06-240/Final Exam Preparation Forum

2006-12-12T20:10:01Z

Wanmike:

06-240/Final Exam Preparation Forum

2006-12-12T20:09:20Z

Wanmike: /* Readings? */

06-240/Final Exam Preparation Forum

2006-12-12T20:07:11Z

Wanmike: /* Sec. 2.4 Lemma p. 101 */

06-240/Final Exam Preparation Forum

2006-12-12T20:06:57Z

Wanmike: /* Sec. 2.4 Lemma p. 101 */

06-240/Final Exam Preparation Forum

2006-12-12T20:06:41Z

Wanmike: /* Sec. 2.4 Lemma p. 101 */

06-240/Final Exam Preparation Forum

2006-12-12T20:06:29Z

Wanmike: /* Sec. 2.4 Lemma p. 101 */

06-240/Final Exam Preparation Forum

2006-12-12T20:06:13Z

Wanmike: /* Sec. 2.4 Lemma p. 101 */

06-240/Final Exam Preparation Forum

2006-12-12T16:14:10Z

Wanmike:

06-240/Final Exam Preparation Forum

2006-12-12T14:40:27Z

Wanmike:

06-240/Final Exam Preparation Forum

2006-12-12T14:38:52Z

Wanmike: /* Sec 3.2 Ex. 19 */

06-240/Final Exam Preparation Forum

2006-12-12T01:32:42Z

Wanmike: /* Readings? */

06-240/Final Exam Preparation Forum

2006-12-12T01:13:53Z

Wanmike:

06-240/Final Exam Preparation Forum

2006-12-12T00:37:16Z

Wanmike:

06-240/Final Exam Preparation Forum

2006-12-11T23:28:12Z

Wanmike: /* Readings? */

06-240/Final Exam Preparation Forum

2006-12-11T20:21:07Z

Wanmike:

06-240/Final Exam Preparation Forum

2006-12-11T17:12:12Z

Wanmike:

Talk:06-240

2006-12-11T05:32:31Z

Wanmike: /* Exam Forum */

If anyone is interested in typesetting the lectures I think they should follow these Wikipedia guidelines: http://en.wikipedia.org/wiki/WP:MSM

==Exam Forum==
I hope you will excuse my intrusion onto the front page, but I thought the link might help increase participation. It will be removed right after the final exam.

Also, my use of "help", leaving it ambiguous as to whether the person clicking on the link would be receiving or giving help, is intentional.

== Modular Arithmetic ==

This was particularly interesting after being introduced to Modular Multiplication tables and seeing some visual patterns with the numbers, such as the in the '1' column where the elements go from 1 to n-1 in Zn and backwards in the 'n-1' column.

After searching around, it seems that people had been able to discover other, more interesting patterns!

Make sure to analyze the tables since they begin from the bottom left corner instead of top left which we saw in class.

http://whistleralley.com/mod/mod25.htm

The following site allows you to see tables up to mod 30.

http://www.cut-the-knot.org/blue/Modulo.shtml

-Richard

Also, notice how in modular multiplication tables for prime numbers, in specific for modulo 5 in the columns and rows for 0 and 5 only 0s appear. The 0s create a sort of frame around a 4x4 square of elements. Specifically all elements within the frame of 0s are between 1 and n-1 and all are non-zero. In the case of the mod 4 table there was a 0 which, as proved in class causes Z4 to fail as a field. There must be something deeper about all those 0s.

== Mistake in the timetable in the main page of 06-240? ==

In the timetable in the main page of 06-240,i.e. http://katlas.math.toronto.edu/drorbn/index.php?title=06-240 , I think there's a mistake about the date of the first test. There, it's written Oct 23th, but we have no class on Oct 23 which is a Monday. I guess the correct date should be Oct 24th as written in the course outline.

-Yanshuai

The dates on the time table are the dates of the Mondays in each week; the header says "week of...". --[[User:Drorbn|Drorbn]] 17:46, 9 October 2006 (EDT)

== TA Office Hours? ==

Do they have any?

Talk:06-240

2006-12-11T05:29:37Z

Wanmike:

If anyone is interested in typesetting the lectures I think they should follow these Wikipedia guidelines: http://en.wikipedia.org/wiki/WP:MSM

==Exam Forum==
I hope you will excuse my intrusion onto the front page, but I thought the link might help increase participation. It will be removed right after the final exam.

== Modular Arithmetic ==

This was particularly interesting after being introduced to Modular Multiplication tables and seeing some visual patterns with the numbers, such as the in the '1' column where the elements go from 1 to n-1 in Zn and backwards in the 'n-1' column.

After searching around, it seems that people had been able to discover other, more interesting patterns!

Make sure to analyze the tables since they begin from the bottom left corner instead of top left which we saw in class.

http://whistleralley.com/mod/mod25.htm

The following site allows you to see tables up to mod 30.

http://www.cut-the-knot.org/blue/Modulo.shtml

-Richard

Also, notice how in modular multiplication tables for prime numbers, in specific for modulo 5 in the columns and rows for 0 and 5 only 0s appear. The 0s create a sort of frame around a 4x4 square of elements. Specifically all elements within the frame of 0s are between 1 and n-1 and all are non-zero. In the case of the mod 4 table there was a 0 which, as proved in class causes Z4 to fail as a field. There must be something deeper about all those 0s.

== Mistake in the timetable in the main page of 06-240? ==

In the timetable in the main page of 06-240,i.e. http://katlas.math.toronto.edu/drorbn/index.php?title=06-240 , I think there's a mistake about the date of the first test. There, it's written Oct 23th, but we have no class on Oct 23 which is a Monday. I guess the correct date should be Oct 24th as written in the course outline.

-Yanshuai

The dates on the time table are the dates of the Mondays in each week; the header says "week of...". --[[User:Drorbn|Drorbn]] 17:46, 9 October 2006 (EDT)

== TA Office Hours? ==

Do they have any?

06-240

2006-12-11T05:28:05Z

Wanmike:

__NOEDITSECTION__
{{06-240/Navigation}}

='''New''': Help with the final exam [[06-240/Final Exam Preparation Forum|here.]]=

==Algebra I==
===Department of Mathematics, University of Toronto, Fall 2006===

{{06-240/Crucial Information}}

===Text===
Freidberg, Insel, Spence. Linear Algebra, 4e. New Jersy: Pearson Education Inc, 2003. 

===Famous Quotes About Math and Mind===
* "The problems that exist in the world today cannot be solved by the level of thinking that created them." by Albert Einstein
* "Do not worry about your difficulties in mathematics, I can assure you mine are still greater." by Albert Einstein
* "The mind, once expanded to the dimensions of larger ideas, never returns to its original size." by Oliver Wendell Holmes, Sr.
* "I don't think necessity is the mother of invention. Invention, in my opinion, arises directly from idleness, possibly also from laziness - to save oneself trouble." by Dame Agatha Christie (1890-1976)
* "A mathematician is a machine for turning coffee into theorems." by Alfréd Rényi, colleague of Paul Erdős.

===Further Resources===

* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]
* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].
* [http://www.math.toronto.edu/murnaghan/courses/mat240/index.html Last year's MAT240 web site].
* [http://www.math.toronto.edu/~megumi/MAT240/240.html Two years ago's MAT240 site].
* [http://www.maths.leeds.ac.uk/~khouston/httlam.html "How to Think Like a Mathematician"] by Kevin Houston.
* [http://www.stonehill.edu/compsci/History_Math/math-read.htm "How to Read Mathematics"] by Shai Simonson and Fernando Gouvea.
* A wide variety of free online mathematics texts is [http://www.math.gatech.edu/~cain/textbooks/onlinebooks.html here]. Want more examples? Get them here. Also, for those in 157 I'd suggest looking at Elias Zakon's Analysis 1 text as a supplement for Spivak.

06-240

2006-12-11T05:27:30Z

Wanmike:

__NOEDITSECTION__
{{06-240/Navigation}}
==Algebra I==
===Department of Mathematics, University of Toronto, Fall 2006===

==='''New''': Help with the final exam [[06-240/Final Exam Preparation Forum|here.]]===

{{06-240/Crucial Information}}

===Text===
Freidberg, Insel, Spence. Linear Algebra, 4e. New Jersy: Pearson Education Inc, 2003. 

===Famous Quotes About Math and Mind===
* "The problems that exist in the world today cannot be solved by the level of thinking that created them." by Albert Einstein
* "Do not worry about your difficulties in mathematics, I can assure you mine are still greater." by Albert Einstein
* "The mind, once expanded to the dimensions of larger ideas, never returns to its original size." by Oliver Wendell Holmes, Sr.
* "I don't think necessity is the mother of invention. Invention, in my opinion, arises directly from idleness, possibly also from laziness - to save oneself trouble." by Dame Agatha Christie (1890-1976)
* "A mathematician is a machine for turning coffee into theorems." by Alfréd Rényi, colleague of Paul Erdős.

===Further Resources===

* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]
* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].
* [http://www.math.toronto.edu/murnaghan/courses/mat240/index.html Last year's MAT240 web site].
* [http://www.math.toronto.edu/~megumi/MAT240/240.html Two years ago's MAT240 site].
* [http://www.maths.leeds.ac.uk/~khouston/httlam.html "How to Think Like a Mathematician"] by Kevin Houston.
* [http://www.stonehill.edu/compsci/History_Math/math-read.htm "How to Read Mathematics"] by Shai Simonson and Fernando Gouvea.
* A wide variety of free online mathematics texts is [http://www.math.gatech.edu/~cain/textbooks/onlinebooks.html here]. Want more examples? Get them here. Also, for those in 157 I'd suggest looking at Elias Zakon's Analysis 1 text as a supplement for Spivak.

06-240

2006-12-11T05:26:26Z

Wanmike:

__NOEDITSECTION__
{{06-240/Navigation}}
==Algebra I==
===Department of Mathematics, University of Toronto, Fall 2006===

'''New''': Help with the final exam [[Final Exam Preparation Forum|here.]]

{{06-240/Crucial Information}}

===Text===
Freidberg, Insel, Spence. Linear Algebra, 4e. New Jersy: Pearson Education Inc, 2003. 

===Famous Quotes About Math and Mind===
* "The problems that exist in the world today cannot be solved by the level of thinking that created them." by Albert Einstein
* "Do not worry about your difficulties in mathematics, I can assure you mine are still greater." by Albert Einstein
* "The mind, once expanded to the dimensions of larger ideas, never returns to its original size." by Oliver Wendell Holmes, Sr.
* "I don't think necessity is the mother of invention. Invention, in my opinion, arises directly from idleness, possibly also from laziness - to save oneself trouble." by Dame Agatha Christie (1890-1976)
* "A mathematician is a machine for turning coffee into theorems." by Alfréd Rényi, colleague of Paul Erdős.

===Further Resources===

* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]
* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].
* [http://www.math.toronto.edu/murnaghan/courses/mat240/index.html Last year's MAT240 web site].
* [http://www.math.toronto.edu/~megumi/MAT240/240.html Two years ago's MAT240 site].
* [http://www.maths.leeds.ac.uk/~khouston/httlam.html "How to Think Like a Mathematician"] by Kevin Houston.
* [http://www.stonehill.edu/compsci/History_Math/math-read.htm "How to Read Mathematics"] by Shai Simonson and Fernando Gouvea.
* A wide variety of free online mathematics texts is [http://www.math.gatech.edu/~cain/textbooks/onlinebooks.html here]. Want more examples? Get them here. Also, for those in 157 I'd suggest looking at Elias Zakon's Analysis 1 text as a supplement for Spivak.

06-240/Final Exam Preparation Forum

2006-12-11T05:23:30Z

Wanmike:

06-240/Final Exam Preparation Forum

2006-12-11T05:21:47Z

Wanmike: /* Exam April/May 2006 #4 */

06-240/Final Exam Preparation Forum

2006-12-11T05:21:06Z

Wanmike: /* Exam April/May 2006 #4 */

06-240/Final Exam Preparation Forum

2006-12-11T05:20:10Z

Wanmike:

06-240/Final Exam Preparation Forum

2006-12-11T05:19:24Z

Wanmike: /* Exam April/May 2006 #4 */

06-240/Final Exam Preparation Forum

2006-12-11T02:43:38Z

Wanmike:

06-240/Final Exam Preparation Forum

2006-12-11T02:42:47Z

Wanmike:

06-240/Final Exam Preparation Forum

2006-12-11T02:42:26Z

Wanmike:

06-240/Final Exam Preparation Forum

2006-12-11T02:39:33Z

Wanmike:

06-240/Final Exam Preparation Forum

2006-12-11T02:33:53Z

Wanmike:

06-240/On The Final Exam

2006-12-11T01:23:48Z

Wanmike:

{{06-240/Navigation}}

'''New:''' Check out the exam preparation forum here: [[06-240/Final Exam Preparation Forum|Final Exam Preparation Forum]].

Our final exam is coming up. It will take place at BN3 - room 3 of the Clara Benson Building, 320 Huron Street (south west of Harbord cross Huron, home of the Faculty of Physical Education and Health) on Wednesday December 13 from 2PM until 5PM. It will consist of 5-6 questions (each may have several parts) on everything that we will have covered in class this semester:
*Fields and vector spaces.
*Spans, independence, replacement and bases.
*Linear transformation, rank, nullity, matrices.
*Row and column reduction and elementary matrices, systems of linear equations.
*Determinants.
*Change of basis and diagonalization.
*Several other "smaller" topics.

As for the style -

*You can expect to be asked to reproduce some proofs that were given in class.
*You can expect some fresh things to prove, though generally not as hard as the previous type of proofs.
*You can expect questions (or parts of questions) that will be identical or nearly identical to questions that were assigned for homework.
*You can expect some calculations (but nothing that will require a calculator).

Basic calculators (not capable of displaying text or sounding speech) will be allowed but will not be necessary. You may wish to bring one nevertheless, as under pressure <math>5+7</math> often comes out to be <math>13</math>.

'''Remember.''' Neatness counts! Organization counts! Language counts! Proofs are best given as short and readable essays; without the English between the formulas one never knows how to interpret those formulas. When you write, say, "<math>v\in V</math>", does it mean "choose <math>v\in V</math>", or "we've just proven that <math>v\in V</math>", or "assume by contradiction that <math>v\in V</math>", or "for every <math>v\in V</math>" or "there exists <math>v\in V</math>"? If you don't say, your reader has no way of knowing. Also remember that long and roundabout solutions of simple problems, full of detours and irrelevant facts, are often an indication that their author didn't quite get the point, even if they are entirely correct. Avoid those!

'''Office hours.''' I ({{Dror}}) will hold two sessions of extended office hours at or near my office (Bahen 6178) before the final:
* On Tuesday December 12th (the day before), 1-4PM.
* On Wednesday December 13th (the day of), 10-12, just for last minute questions.
I would have loved to give more, but long before the final was scheduled I was asked to organize a [[CMS Winter 2006 Session on Knot Homologies|session]] in a Canadian Mathematical Society conference on Saturday through Monday right before our exam. So unfortunately I will be completely unreachable on these three days.

06-240/Final Exam Preparation Forum

2006-12-10T23:01:48Z

Wanmike:

06-240/Final Exam Preparation Forum

2006-12-10T23:01:37Z

Wanmike:

Template:06-240/Navigation

2006-12-10T21:55:32Z

Wanmike:

{| cellpadding="0" cellspacing="0" style="clear: right; float: right"
|- align=right
|<div class="NavFrame"><div class="NavHead">[[06-240]]/[[Template:06-240/Navigation|Navigation Panel]]  </div>
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{| border="1px" cellpadding="1" cellspacing="0" width="220" style="margin: 0 0 1em 0.5em; font-size: small"
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!#
!Week of...
!Notes and Links
|-
|align=center|1
|Sep 11
|[[06-240/About This Class|About]], [[06-240/Classnotes For Tuesday, September 12|Tue]], [[06-240/Homework Assignment 1|HW1]], [[06-240/Putnam Competition|Putnam]], [[06-240/Classnotes for Thursday, September 14|Thu]]
|-
|align=center|2
|Sep 18
|[[06-240/Classnotes For Tuesday, September 19|Tue]], [[06-240/Homework Assignment 2|HW2]], [[06-240/Classnotes For Thursday, September 21|Thu]]
|-
|align=center|3
|Sep 25
|[[06-240/Classnotes For Tuesday September 26|Tue]], [[06-240/Homework Assignment 3|HW3]], [[06-240/Class Photo|Photo]], [[06-240/Classnotes For Thursday, September 28|Thu]]
|-
|align=center|4
|Oct 2
|[[06-240/Classnotes For Tuesday October 3|Tue]], [[06-240/Homework Assignment 4|HW4]], [[06-240/Classnotes For Thursday October 5|Thu]]
|-
|align=center|5
|Oct 9
|[[06-240/Classnotes For Tuesday October 10|Tue]], [[06-240/Homework Assignment 5|HW5]], [[06-240/Classnotes For Thursday October 12|Thu]]
|-
|align=center|6
|Oct 16
|[[06-240/Linear Algebra - Why We Care|Why?]], [http://en.wikipedia.org/wiki/Isomorphism Iso], [[06-240/Classnotes For Tuesday October 17|Tue]], [[06-240/Classnotes For Thursday October 19|Thu]]
|-
|align=center|7
|Oct 23
|[[06-240/Term Test|Term Test]], [[06-240/Classnotes For Thursday October 26|Thu (double)]]
|-
|align=center|8
|Oct 30
|[[06-240/Classnotes For Tuesday October 31|Tue]], [[06-240/Homework Assignment 6|HW6]], [[06-240/Classnotes For Thursday November 2|Thu]]
|-
|align=center|9
|Nov 6
|[[06-240/Classnotes For Tuesday November 7|Tue]], [[06-240/Homework Assignment 7|HW7]], [[06-240/Classnotes For Thursday November 9|Thu]]
|-
|align=center|10
|Nov 13
|[[06-240/Classnotes For Tuesday November 14|Tue]], [[06-240/Homework Assignment 8|HW8]], [[06-240/Classnotes For Thursday November 16|Thu]]
|-
|align=center|11
|Nov 20
|[[06-240/Classnotes For Tuesday November 21|Tue]], [[06-240/Homework Assignment 9|HW9]], [[06-240/Classnotes For Thursday November 23|Thu]]
|-
|align=center|12
|Nov 27
|[[06-240/Classnotes For Tuesday November 28|Tue]], [[06-240/Homework Assignment 10|HW10]], [[06-240/Classnotes For Thursday November 30|Thu]]
|-
|align=center|13
|Dec 4
|[[06-240/On The Final Exam|On the final]], [[06-240/Classnotes For Tuesday December 5|Tue]], [[06-240/Classnotes For Thursday December 7|Thu]]
|-
|align=center|F
|Dec 11
|Final: Dec 13 2-5PM at BN3, [[06-240/Final Exam Preparation Forum|Exam Forum]]
|-
|colspan=3 align=center|[[06-240/Register of Good Deeds|Register of Good Deeds]]
|-
|colspan=3 align=center|[[Image:06-240-ClassPhoto.jpg|180px]] [[06-240/Class Photo|Add your name / see who's in!]]
|-
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|}
</div></div>
|}

06-240/Final Exam Preparation Forum

2006-12-10T21:45:33Z

Wanmike:

06-240/On The Final Exam

2006-12-10T21:43:23Z

Wanmike:

{{06-240/Navigation}}

'''New:''' Check out the exam preparation forum [[06-240/Final Exam Preparation Forum|here]].

Our final exam is coming up. It will take place at BN3 - room 3 of the Clara Benson Building, 320 Huron Street (south west of Harbord cross Huron, home of the Faculty of Physical Education and Health) on Wednesday December 13 from 2PM until 5PM. It will consist of 5-6 questions (each may have several parts) on everything that we will have covered in class this semester:
*Fields and vector spaces.
*Spans, independence, replacement and bases.
*Linear transformation, rank, nullity, matrices.
*Row and column reduction and elementary matrices, systems of linear equations.
*Determinants.
*Change of basis and diagonalization.
*Several other "smaller" topics.

As for the style -

*You can expect to be asked to reproduce some proofs that were given in class.
*You can expect some fresh things to prove, though generally not as hard as the previous type of proofs.
*You can expect questions (or parts of questions) that will be identical or nearly identical to questions that were assigned for homework.
*You can expect some calculations (but nothing that will require a calculator).

Basic calculators (not capable of displaying text or sounding speech) will be allowed but will not be necessary. You may wish to bring one nevertheless, as under pressure <math>5+7</math> often comes out to be <math>13</math>.

'''Remember.''' Neatness counts! Organization counts! Language counts! Proofs are best given as short and readable essays; without the English between the formulas one never knows how to interpret those formulas. When you write, say, "<math>v\in V</math>", does it mean "choose <math>v\in V</math>", or "we've just proven that <math>v\in V</math>", or "assume by contradiction that <math>v\in V</math>", or "for every <math>v\in V</math>" or "there exists <math>v\in V</math>"? If you don't say, your reader has no way of knowing. Also remember that long and roundabout solutions of simple problems, full of detours and irrelevant facts, are often an indication that their author didn't quite get the point, even if they are entirely correct. Avoid those!

'''Office hours.''' I ({{Dror}}) will hold two sessions of extended office hours at or near my office (Bahen 6178) before the final:
* On Tuesday December 12th (the day before), 1-4PM.
* On Wednesday December 13th (the day of), 10-12, just for last minute questions.
I would have loved to give more, but long before the final was scheduled I was asked to organize a [[CMS Winter 2006 Session on Knot Homologies|session]] in a Canadian Mathematical Society conference on Saturday through Monday right before our exam. So unfortunately I will be completely unreachable on these three days.