Drorbn - User contributions [en]

File:Testsoln2.pdf

2014-12-08T19:18:50Z

Boyang.wu:

File:Testsoln1.pdf

2014-12-08T19:18:17Z

Boyang.wu:

14-240/Term Test

2014-12-08T19:17:15Z

Boyang.wu: /* The Results */

{{14-240/Navigation}}

Our Term Test took place at '''HS 610''' (Health Sciences 610) on Tuesday October 21, 1:10PM-3PM. The results are available on the UofT Portal.

'''The Test:''' {{pensieve link|Classes/14-240/TT-240.pdf|TT-240.pdf}}.

==The Results==
Excluding some exceptions, 121 students took the test. Before appeals, the average grade was approximately 74 and the standard deviation was approximately 25. The full list of marks (before appeals) was as follows:

<blockquote>
100 100 100 100 100 100 100 100 100 100 100 100 99 99 98 98 98 98 98 97 97 97 97 97 97 97 97 97 97 96 96 96 96 95 95 95 95 94 94 94 93 93 93 92 92 92 91 91 90 90 90 89 88 88 88 88 88 88 87 87 86 82 82 81 80 79 79 76 74 73 73 71 70 70 70 70 69 69 69 68 62 61 60 60 60 60 59 59 58 58 58 58 54 54 53 53 50 50 50 49 47 45 43 42 41 41 41 40 40 40 39 37 36 35 25 21 18 16 15 8 4
</blockquote>

The results are quite similar to what I expected them to be. The easiest questions (on average) were the computational ones, the hardest were the ones involving proofs.

How should you read your grade?

* If you got 100 you should pat yourself on your shoulder and feel good.
* If you got something like 95, you're doing great. You made a few relatively minor mistakes; find out what they are and try to avoid them next time.
* If you got something like 85 you're doing fine but you did miss something significant, probably more than just a minor thing. Figure out what it was and make a plan to fix the problem for next time.
* If you got something like 65 you should be concerned. You are still in position to improve greatly and get an excellent grade at the end, but what you missed is quite significant and you are at the risk of finding yourself far behind. You must analyze what happened - perhaps it was a minor mishap, but more likely you misunderstood something major or something major is missing in your background. Find out what it is and try to come up with a realistic strategy to overcome the difficulty!
* If you got something like 45, most likely you are not gaining much from this class and you should consider dropping it, unless you are convinced that you fully understand the cause of your difficulty (you were very sick, you really couldn't study at all for the two weeks before the exam because of some unusual circumstances, something like that) and you feel confident you have a fix for next time. If you do decide to drop the class, don't feel too bad about it. It is the hardest first year algebra class at UofT and of the thousands of students taking math here, very few come with sufficient preparation to do well in it.

Note that problems with writing are problems, period. Perhaps you got a low grade but you feel you know the material enough for a high grade only you didn't write everything you know or you didn't it write well enough or the silly graders simply didn't get what you wrote (and it isn't a simple misunderstanding - see "appeals" below). If this describes you, don't underestimate your problem. If you don't process and resolve it, it is likely to recur.

====Appeals.====
Remember! We try hard yet grading is a difficult process and mistakes '''always''' happen - solutions get misread, parts are forgotten, grades are not added up correctly. You '''must''' read your exam and make sure that you understand how it was graded. If you disagree with anything, don't hesitate to complain! (Though first consider very carefully the possibility that the mistake is actually yours). Your first stop should be the person who graded the problem in question, and only if you can't agree with him you should appeal to {{Dror}}.

{{Dror}} graded problem number 1 and did the data entry. Boris Lishak graded problems 2 and 4, and Nikita Nikolaev graded 3 and 5.

The deadline to start the appeal process is Wednesday November 5 at noon. Once you've started the process by talking to {{Dror}} or to Boris or to Nikita, it ends when a final decision is made, with no deadline.

{{14-240:Dror/Students Divider}}

I've attempted to make a list of all defined terms and theorems (currently from the textbook chapters 1.1 - 1.6 and appendices A-D, I might add more later) in this [[http://pastebin.com/igPXqtyk latex],
[http://drorbn.net/index.php?title=File:Theorems-defined-terms-term-test.pdf pdf]] file. Hopefully someone will find this useful, and I hope this is an appropriate place to post this.

==Scanned Lecture Notes by [[User Boyang.wu|Boyang.wu]]==

[[File:Testsoln1.pdf]]
[[File:Testsoln2.pdf]]

File:Testsoln2pdf.pdf

2014-12-08T19:16:24Z

Boyang.wu:

File:Testsoln1pdf.pdf

2014-12-08T19:16:17Z

Boyang.wu:

14-240/Tutorial-October14

2014-12-08T19:08:52Z

Boyang.wu: /* Nikita */

{{14-240/Navigation}}

==Boris==

====Elementary and (Not So Elementary) Errors in Homework====

'''(1) Bad Notation'''

Consider these three matrices:
::::::::<math>
M_1 =
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix},
M_2 =
\begin{pmatrix}
0 & 0 \\
0 & 1 \\
\end{pmatrix},
M_3 =
\begin{pmatrix}
0 & 1 \\
1 & 0 \\
\end{pmatrix}
</math>

We want to equate <math>span(M_1, M_2, M_3)</math> to the set of all symmetric <math>2 \times 2</math> matrices. Here is the wrong way to write this:

::::::::::::<math>
span(M_1, M_2, M_3) =
\begin{pmatrix}
a & b \\
b & c \\
\end{pmatrix}
</math>.

Firstly, <math>span(M_1, M_2, M_2)</math> is the set of all linear combinations of <math>M_1, M_2, M_3</math>. To equate it to a single symmetric <math>2 \times 2</math> matrix makes no sense. Secondly, the elements <math>a, b, c, d</math> are undefined. What are they suppose to represent? Rational numbers? Real numbers? Members of the field of two elements? The following way of writing erases those issues:

:::::<math>
span(M_1, M_2, M_3) = \{
\begin{pmatrix}
a & b \\
b & c \\
\end{pmatrix}
:a, b, c \in F \}
</math> where <math>F</math> is an arbitrary field.

'''(2) Algorithm vs. Proof (Boris's Section Only)'''

When solving a problem that requires a solution to a linear equation, it is not always obvious which of the following you should show:
:a) An algorithm for finding the solution
:b) A proof that a solution is correct
If the problem asks to solve a linear equation, then just show (a). Otherwise, consider problems such as this:

:Determine if the vector <math>(-2, 2, 2)</math> is a linear combination of the vectors <math>(- (1, 2, -1), (-3, -3, 3)</math> in <math>R^3</math>.

Show both (a) and (b) to be on the safe side.

====Problem 5h) on Page 34 in Homework 3 for all Fields====

For an arbitrary field <math>F</math>, determine if the matrix
<math>
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
\end{pmatrix}
</math>
is in span
<math>S=\{
\begin{pmatrix}
1 & 0 \\
-1 & 0 \\
\end{pmatrix},
\begin{pmatrix}
0 & 1 \\
0 & 1 \\
\end{pmatrix},
\begin{pmatrix}
1 & 1 \\
0 & 0 \\
\end{pmatrix}
\}
</math>.

'''Proof:'''

We show that
<math>
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
\end{pmatrix}
\in span(S) \iff char(F)=2</math>.

:We show that <math>char(F)=2 \implies
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
\end{pmatrix}
\in span(S)
</math>.
::Assume that <math>char(F)=2</math>.

::Let <math>c_1=0, c_2=1, c_3=1</math>.

::Then <math>c_1, c_2, c_3 \in F</math>.

::Then <math>
c_1
\begin{pmatrix}
1 & 0 \\
-1 & 0 \\
\end{pmatrix}
+ c_2
\begin{pmatrix}
0 & 1 \\
0 & 1 \\
\end{pmatrix}
+ c_3
\begin{pmatrix}
1 & 1 \\
0 & 0 \\
\end{pmatrix}
=
\begin{pmatrix}
1 & 2 \\
0 & 1 \\
\end{pmatrix}
</math>.

::Since <math>char(F)=2</math> and the entries of the matrix are from <math>F</math>, then <math>0=2</math>.

::Then <math>
\begin{pmatrix}
1 & 2 \\
0 & 1 \\
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
\end{pmatrix}
\in span(S)
</math>.

::Then <math>
char(F)=2 \implies
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
\end{pmatrix}
\in span(S)
</math>.

:We show that <math>char(F) \neq 2 \implies
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
\end{pmatrix}
\notin span(S)
</math>.
::Assume to the contrary that <math>
char(F) \neq 2 \and
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
\end{pmatrix}
\in span(S)
</math>.

::Then <math>\exists c_1, c_2, c_3 \in F,
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
\end{pmatrix}
=c_1
\begin{pmatrix}
1 & 0 \\
-1 & 0 \\
\end{pmatrix}
+c_2
\begin{pmatrix}
0 & 1 \\
0 & 1 \\
\end{pmatrix}
+c_3
\begin{pmatrix}
1 & 1 \\
0 & 0 \\
\end{pmatrix}
</math>.

::Then this system of linear equations has a solution:

:::<math>(11)c_1+c_3=1</math>

:::<math>(21)-c_1=0</math>

:::<math>(12)c_2+c_3=0</math>

:::<math>(22)c_2=1</math>.

::When solving this system, we see that it has no solution.

::This contradicts the assumption that it has a solution.

::Then <math>
char(F) \neq2 \implies
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
\end{pmatrix}
\notin span(S)
</math>.

:Then <math>
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
\end{pmatrix}
\in span(S) \iff char(F)=2</math>. ''Q.E.D.''

====A Field Problem====

Find the solution to <math>x^2 = -2</math> in <math>Z_{11}</math>.

Note that a polynomial of <math>n</math> degree has at most <math>n</math> solutions.

''Algorithm:''

We find the solution to <math>x^2 = -2</math> in <math>Z_{11}</math>.

:Since in <math>Z_{11}, -2 = 9</math>, then <math>x^2 = 9</math>.

:Since <math>-9</math> is additive inverse of <math>9</math>, then <math>x^2 - 9 = 9 - 9 = 0</math>.

:By the result that we proved in ''Question 2 of Homework 1'', then <math>(x^2 - 9) = (x - 3)(x + 3) = 0</math>.

:Then <math>x = \pm 3</math> are the solutions.

====A Dimension Problem====

Let <math>W_1, W_2</math> be subspaces of a finite dimensional vector space <math>V</math> over a field <math>F</math> where <math>W_1 \cap W_2 = \{0\}</math>. Then

<math>dim(span(W_1 \cup W_2)) = dim(W_1) + dim(W_2)</math>.

''Proof:''

We show that <math>dim(span(W_1 \cup W_2)) = dim(W_1) + dim(W_2)</math>.

:Since <math>V</math> is finite dimensional, then <math>W_1, W_2</math> are finite dimensional.

:Then we can let <math>B_1 = \{u_1, u_2, u_3, ..., u_m\}</math> be a basis of <math>W_1</math> and <math>B_2 = \{v_1, v_2, v_3, ..., v_n\}</math> be a basis of <math>W_2</math>.

:We show that <math>B_1 \cup B_2</math> is a basis of <math>span(W_1 \cup W_2)</math>.

::We show that <math>span(B_1 \cup B_2) = span(W_1 \cup W_2)</math>.

:::We show that <math>span(B_1 \cup B_2) \subset span(W_1 \cup W_2)</math>.

::::Since <math>(B_1 \cup B_2) \subset (W_1 \cup W_2)</math>, then <math>span(B_1 \cup B_2) \subset span(W_1 \cup W_2)</math>.

:::We show that <math>span(W_1 \cup W_2) \subset span(B_1 \cup B_2)</math>.

::::Since <math>B_1 \subset (B_1 \cup B_2)</math>, then <math>span(B_1) = W_1 \subset span(B_1 \cup B_2)</math>.

::::Since <math>B_2 \subset (B_1 \cup B_2)</math>, then <math>span(B_2) = W_2 \subset span(B_1 \cup B_2)</math>.

::::Then <math>(W_1 \cup W_2) \subset span(B_1 \cup B_2)</math> and <math>span(W_1 \cup W_2) \subset span(B_1 \cup B_2)</math>.

:::Then <math>span(B_1 \cup B_2) = span(W_1 \cup W_2)</math>.

::We show that <math>B_1 \cup B_2</math> is linearly independent.

:::Let <math>\displaystyle\sum_{i=1}^{m} b_iu_i + \displaystyle\sum_{j=1}^{n} c_jv_j = 0</math> where <math>b_i, c_j \in F</math>.

:::Then <math>\displaystyle\sum_{i=1}^{m} b_iu_i = \displaystyle\sum_{j=1}^{n} (-c_j)v_j</math>.

:::Since <math>W_1 \cap W_2 = \{ 0 \}</math>, then <math>\displaystyle\sum_{i=1}^{m} b_iu_i = \displaystyle\sum_{j=1}^{n} (-c_j)v_j = 0</math>.

:::Since <math>B_1, B_2</math> are linearly independent, then <math>b_i = (-c_j) = 0</math>.

:::For <math>\displaystyle\sum_{i=1}^{m} b_iu_i + \displaystyle\sum_{j=1}^{n} c_jv_j = 0</math>, then <math>b_i = c_j = 0</math>.

:::Then <math>B_1 \cup B_2</math> is linearly independent.

::Then <math>B_1 \cup B_2</math> is a basis of <math>span(W_1 \cup W_2)</math>.

:Since <math>\left| B_1 \cup B_2 \right| = \left| B_1 \right| + \left| B_2 \right|</math>, then <math>dim(span(W_1 \cup W_2)) = dim(W_1) + dim(W_2)</math>. ''Q.E.D.''

==Nikita==
[[File:1014.240.pdf]]

==Scanned Tutorial Notes by [[User Boyang.wu|Boyang.wu]]==
[[File:Tut6.pdf]]

14-240/Homework Assignment 8

2014-12-08T19:08:06Z

Boyang.wu:

{{14-240/Navigation}}
This assignment is due at the tutorials on Tuesday November 25 or at the appropriate mailboxes at the Math Aid Centre, SS 1071, by Thursday November 27 at 5PM. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

'''Reread''' 3.1 and 3.2 in our textbook, and '''read''' sections 3.3 and 3.4. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, '''preread''' section 2.5, just to get a feel for the future.

'''Solve''' problems 1, 2, 3, 8 and 10 on pages 179-181 and problems 1, 2abe, 2cdfghij and 5 on pages 195-196 but submit only your solutions of the underlined problems.

==Scanned Assignment Solutions by [[User Boyang.wu|Boyang.wu]]==
[[File:A81.pdf]]
[[File:A82.pdf]]
[[File:A83.pdf]]

14-240/Homework Assignment 7

2014-12-08T19:07:41Z

Boyang.wu:

{{14-240/Navigation}}
This assignment is due at the tutorials on Tuesday November 11 or at the appropriate mailboxes at the Math Aid Centre, SS 1071, by Thursday November 20 at 5PM. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

'''Task 1.''' '''Read''' sections 2.4, 3.1 and 3.2 in our textbook. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, '''preread''' sections 2.5 and 3.3-3.4, just to get a feel for the future.

'''Task 2.''' '''Solve''' problems 1, 2, 3a, and 3b on pages 96-97, problems 1, 2, 4 and 9 on pages 106-107, problems 1, 2, and 9 on pages 151-152, and problems 1, 2aceg, 2bdf, 7, 19, and 21 on pages 165-168, yet submit only your solutions of the underlined problems.

==Scanned Assignment Solutions by [[User Boyang.wu|Boyang.wu]]==
[[File:A71.pdf]]
[[File:A72.pdf]]
[[File:A73.pdf]]

14-240/Homework Assignment 6

2014-12-08T19:07:09Z

Boyang.wu:

{{14-240/Navigation}}
This assignment is due at the tutorials on Tuesday November 4 or at the appropriate mailboxes at the Math Aid Centre, SS 1071, by Thursday November 6 at 5PM. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

'''Task 1.''' '''Read''' sections 2.1 through 2.3 in our textbook. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, '''preread''' sections 2.4 and 2.5, just to get a feel for the future.

'''Task 2.''' '''Solve''' problems 20, 21, 24, 25, 26, and 27 on pages 76-77 and problems 1, 5a, 5b, 5c-g, 9, 15 and 16 on pages 84-86, but submit only your solutions of the underlined problems.

'''Hint for Problem 16 on Page 86.''' Start with the same bases as in the proof of rank-nullity, add just a bit, and what is the resulting matrix?

'''Just for fun.''' Do a web search for the words "OpenGL" and "matrix" (or "matrices") and find out why graphics accelerators in PCs and game consoles do <math>4\times 4</math> (!) matrix multiplications ''in hardware''. (The full explanation is quite involved; perhaps you should just aim to get a taste of it).

[[Image:06-240-Mario.jpg|thumb|center|176px|A famed creature that owes its mobility to 4x4 matrices.]]

{{14-240:Dror/Students Divider}}

==Scanned Assignment Solutions by [[User Boyang.wu|Boyang.wu]]==
[[File:A61.pdf]]
[[File:A62.pdf]]
[[File:A63.pdf]]
[[File:A64.pdf]]

14-240/Homework Assignment 5

2014-12-08T19:06:35Z

Boyang.wu:

{{14-240/Navigation}}
This assignment is due at the tutorials on Tuesday October 28 or at the appropriate mailboxes at the Math Aid Centre, SS 1071, by Thursday October 30 at 5PM. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

'''Task 1.''' Read/reread sections 1.6, 1.7, and 2.1 in our textbook. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Remember that your prof. thinks that section 1.7 is useless fun. Also, '''preread''' the rest of chapter 2, just to get a feel for the future.

'''Task 2.''' Solve problems 17, 18, 20, 23, 25, 26, 28, 29a, and 29b on pages 56-57 and problems 1, 2, 5, 13, 17 and 18 on pages 74-76, but submit only your solutions of the underlined problems.

'''Just for fun.''' Decide if the vectors <math>\begin{pmatrix}8\\-3\end{pmatrix}</math> and <math>\begin{pmatrix}5\\-2\end{pmatrix}</math> are linearly dependent.

<center>'''How Can This Be?'''</center>
[[Image:How Can This Be.png|thumb|350px|center|Two congruent triangles are assembled using congruent pieces, yet one is bigger than the other]]

==Scanned Assignment Solutions by [[User Boyang.wu|Boyang.wu]]==
[[File:A51.pdf]]
[[File:A52.pdf]]
[[File:A53.pdf]]

14-240/Homework Assignment 4

2014-12-08T19:06:10Z

Boyang.wu:

{{14-240/Navigation}}

This assignment is due at the tutorials on Tuesday October 14 or at the appropriate mailboxes at the Math Aid Centre, SS 1071, by Thursday October 16 at 5PM. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

'''Task 0.''' Add your name to the [[14-240/Class Photo|Class Photo]] page!

'''Task 1.''' Read sections 1.5 through 1.7 in our textbook. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read.

'''Task 2.''' Solve problems 3, 8, 9, 10, and 11 on pages 41-42, but submit only your solutions of problems 8, 9, and 11.

'''Task 3.''' Solve problems 1, 2, 4, 5, 9, 12, 13, and 16 on page 53-56, but submit only your solutions of problems 4, 5, 9, and 12.

{|
|-
|'''Just for Fun (1).'''
* Take a large integer and write it in base 10. Cut away the "singles" digit, double it and subtract the result from the remaining digits. Repeat the process until the number you have left is small. Prove that the number you started from is divisible by 7 iff the resulting number is divisible by 7. Thus the example on the right shows that 86415 is divisible by 7 as 0 is divisible by 7.
* Find a similar criterion for divisibility by 17 and for all other divisibilities and indivisibilities.
* Note that the word "indivisibilities" has the largest number of repetitions of a single letter among all words in the English language (7 i's). I've known this fact for many years yet here's a semi-legitimate use for that word! (It is tied with the word honorificabilitudinitatibus for seven 'i's. You can read more about it here: http://en.wikipedia.org/wiki/Honorificabilitudinitatibus)
|
8641<s>5</s>
10
----
863<s>1</s>
2
---
86<s>1</s>
2
--
8<s>4</s>
8
-
0
|}

'''Just for Fun (2).''' Is there a problem with the following inductive proof that all horses are of the same color?

We assert that in all sets with precisely <math>n</math> horses, all horses are of the same color. For <math>n=1</math>, this is obvious: it is clear that in a set with just one horse, all horses are of the same color. Now assume our assertion is true for all sets with <math>n-1</math> horses, and let us be given a set with <math>n</math> horses in it. By the inductive assumption, the first <math>n-1</math> of those are of the same color and also the last <math>n-1</math> of those. Hence they are all of the same color as illustrated below:

{{Equation*|<math>(H,[H,\ldots,H),H]</math>}}

(The horses surrounded by round brackets <math>(\cdots)</math> are all of the same color. The horses surrounded by square brackets <math>[\cdots]</math> are all of the same color. Therefore the first and the last horses have the same color as the ones in the middle group, and hence all horses are of the same color.)

==Scanned Assignment Solutions by [[User Boyang.wu|Boyang.wu]]==
[[File:A41.pdf]]
[[File:A42.pdf]]

14-240/Homework Assignment 3

2014-12-08T19:05:38Z

Boyang.wu:

{{14-240/Navigation}}

Homework assignment 3 is due at the tutorials on Tuesday October 7 or at the appropriate mailboxes at the Math Aid Centre, SS 1071, by Thursday October 9 at 5PM. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

'''Task 0.''' Add your name to the [[14-240/Class Photo|Class Photo]] page!

'''Task 1.''' Read sections 1.4-1.6 in our textbook. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read.

'''Task 2.'''
* Solve problems 1-5 on pages 32-34, but submit only the last part of each problem.
* Solve problems 6-10 on page 34, but submit only your solution of problem 10.

==Scanned Assignment Solutions by [[User Boyang.wu|Boyang.wu]]==
[[File:A31.pdf]]
[[File:A32.pdf]]

14-240/Homework Assignment 2

2014-12-08T19:05:04Z

Boyang.wu:

{{14-240/Navigation}}
This assignment is due at the tutorials on Tuesday September 30, or at the Math Aid Centre, Sydney Smith room 1071, at the appropriately labeled mailboxes, by Friday October 3 at 5PM. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

Read sections 1.1 through 1.4 in our textbook, and solve the following problems:

* Problems 3a and 3bcd on page 6, problems 1, 7, 18, 19 and 21 on pages 12-16 and problems 8, 9, 11 and 19 on pages 20-21. You need to submit only the underlined problems.

* Note that the numbers <math>1^6-1=0</math>, <math>2^6-1=63</math>, <math>3^6-1=728</math>, <math>4^6-1=4,095</math>, <math>5^6-1=15,624</math> and <math>6^6-1=46,655</math> are all divisible by <math>7</math>. The following four part exercise explains that this is not a coincidence. But first, let <math>p</math> be some odd prime number and let <math>{\mathbb F}_p</math> be the field with p elements as defined in class.
*# Prove that the product <math>b:=1\cdot 2\cdot\ldots\cdot(p-2)\cdot(p-1)</math> is a non-zero element of <math>{\mathbb F}_p</math>.
*# Let <math>a</math> be a non-zero element of <math>{\mathbb F}_p</math>. Prove that the sets <math>\{1,2,\ldots,(p-1)\}</math> and <math>\{1a,2a,\ldots,(p-1)a\}</math> are the same (though their elements may be listed here in a different order).
*# With <math>a</math> and <math>b</math> as in the previous two parts, show that <math>ba^{p-1}=b</math> in <math>{\mathbb F}_p</math>, and therefore <math>a^{p-1}=1</math> in <math>{\mathbb F}_p</math>.
*# How does this explain the fact that <math>4^6-1</math> is divisible by <math>7</math>?
:You don't need to submit this exercise at all, but you will learn a lot by doing it!

* After September 24, add your name to the [[14-240/Class Photo|Class Photo]] page!

==Scanned Assignment Solutions by [[User Boyang.wu|Boyang.wu]]==
[[File:A21.pdf]]
[[File:A22.pdf]]
[[File:A23.pdf]]
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14-240/Homework Assignment 1

2014-12-08T19:04:33Z

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This assignment is due at the tutorials on Tuesday September 23. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

Read appendices A through D in our textbook (with higher attention to C and D), and solve the following problems:

# Suppose <math>a</math> and <math>b</math> are nonzero elements of a field <math>F</math>. Using only the field axioms, prove that <math>a^{-1}b^{-1}</math> is a multiplicative inverse of <math>ab</math>. State which axioms are used in your proof.
# Prove that if <math>a</math> and <math>b</math> are elements of a field <math>F</math>, then <math>a^2=b^2</math> if and only if <math>a=b</math> or <math>a=-b</math>.
# Let <math>F_4=\{0,1,a,b\}</math> be a field containing 4 elements. Assume that <math>1+1=0</math>. Prove that <math>b=a^{-1}=a^2=a+1</math>. (''Hint:'' For example, for the first equality, show that <math>a\cdot b</math> cannot equal <math>0</math>, <math>a</math>, or <math>b</math>.)
# Write the following complex numbers in the form <math>a+ib</math>, with <math>a,b\in{\mathbb R}</math>:
## <math>\frac{1}{2i}+\frac{-2i}{5-i}</math>.
## <math>(1+i)^5</math>.
#
## Prove that the set <math>F_1=\{a+b\sqrt{3}:a,b\in{\mathbb Q}\}</math> (endowed with the addition and multiplication inherited from <math>{\mathbb R}</math>) is a field.
## Is the set <math>F_2=\{a+b\sqrt{3}:a,b\in{\mathbb Z}\}</math> (with the same addition and multiplication) also a field?

==Scanned Assignment Solutions by [[User Boyang.wu|Boyang.wu]]==
[[File:A11.pdf]]
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14-240/Homework Assignment 1

2014-12-08T19:04:19Z

Boyang.wu: /* Scanned Assignment Solution by Boyang.wu */

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This assignment is due at the tutorials on Tuesday September 23. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

Read appendices A through D in our textbook (with higher attention to C and D), and solve the following problems:

# Suppose <math>a</math> and <math>b</math> are nonzero elements of a field <math>F</math>. Using only the field axioms, prove that <math>a^{-1}b^{-1}</math> is a multiplicative inverse of <math>ab</math>. State which axioms are used in your proof.
# Prove that if <math>a</math> and <math>b</math> are elements of a field <math>F</math>, then <math>a^2=b^2</math> if and only if <math>a=b</math> or <math>a=-b</math>.
# Let <math>F_4=\{0,1,a,b\}</math> be a field containing 4 elements. Assume that <math>1+1=0</math>. Prove that <math>b=a^{-1}=a^2=a+1</math>. (''Hint:'' For example, for the first equality, show that <math>a\cdot b</math> cannot equal <math>0</math>, <math>a</math>, or <math>b</math>.)
# Write the following complex numbers in the form <math>a+ib</math>, with <math>a,b\in{\mathbb R}</math>:
## <math>\frac{1}{2i}+\frac{-2i}{5-i}</math>.
## <math>(1+i)^5</math>.
#
## Prove that the set <math>F_1=\{a+b\sqrt{3}:a,b\in{\mathbb Q}\}</math> (endowed with the addition and multiplication inherited from <math>{\mathbb R}</math>) is a field.
## Is the set <math>F_2=\{a+b\sqrt{3}:a,b\in{\mathbb Z}\}</math> (with the same addition and multiplication) also a field?

==Scanned Assignment Solution by [[User Boyang.wu|Boyang.wu]]==
[[File:A11.pdf]]
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Boyang.wu:

14-240/Homework Assignment 1

2014-12-08T18:18:08Z

Boyang.wu:

14-240/Tutorial-December 2

2014-12-08T18:17:32Z

Boyang.wu: /* Scanned Lecture Notes by Boyang.wu */

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==Boris==

====Theorem====

Let <math>A</math> be a <math>n \times n</math> matrix and <math>B</math> be the matrix <math>A</math> with two rows interchanged. Then <math>det(A) = -det(B)</math>. Boris decided to prove the following lemma first:

=====Lemma 1=====

Let <math>A</math> be a <math>n \times n</math> matrix and <math>B</math> be the matrix <math>A</math> with two '''adjacent''' rows interchanged. Then <math>det(A) = -det(B)</math>.

All we need to show is that <math>det(A) + det(B) = 0</math>. Assume that <math>B</math> is the matrix <math>A</math> with rows <math>i, i + 1</math> of <math>A</math> interchanged. Since the determinant of a matrix with two identical rows is <math>0</math>, then:

:::::::<math>det(A) + det(B) =</math>

:::::::<math>det\begin{pmatrix}...\\A_i\\A_{i + 1}\\...\end{pmatrix} + det\begin{pmatrix}...\\A_{i + 1}\\A_i\\...\end{pmatrix} =</math>

:::::::<math>det\begin{pmatrix}...\\A_i\\A_i\\...\end{pmatrix} + det\begin{pmatrix}...\\A_i\\A_{i + 1}\\...\end{pmatrix} + det\begin{pmatrix}...\\A_{i + 1}\\A_{i + 1}\\...\end{pmatrix} + det\begin{pmatrix}...\\A_{i + 1}\\A_i\\...\end{pmatrix}</math>.

Since the determinant is linear in each row, then we continue where we left off:

:::::::<math>det(A) + det(B) =</math>

:::::::<math>det\begin{pmatrix}...\\A_i\\A_i\\...\end{pmatrix} + det\begin{pmatrix}...\\A_i\\A_{i + 1}\\...\end{pmatrix} + det\begin{pmatrix}...\\A_{i + 1}\\A_{i + 1}\\...\end{pmatrix} + det\begin{pmatrix}...\\A_{i + 1}\\A_i\\...\end{pmatrix} = </math>

:::::::<math>det\begin{pmatrix}...\\A_i\\A_i + A_{i + 1}\\...\end{pmatrix} + det\begin{pmatrix}...\\A_{i + 1}\\A_i + A_{i + 1}\\...\end{pmatrix} = det\begin{pmatrix}...\\A_i + A_{i + 1}\\A_i + A_{i + 1}\\...\end{pmatrix} = 0</math>.

Then <math>det(A) + det(B) = 0</math> and <math>det(A) = -det(B)</math>. The proof of the lemma is complete.

For the proof of the theorem, assume that <math>B</math> is the matrix <math>A</math> with rows <math>i, j</math> of <math>A</math> interchanged and <math>i \neq j</math>. By '''Lemma 1''', we have the following:

:::::::<math>det(A) =</math>

:::::::<math>(-1)det\begin{pmatrix}...\\A_{i + 1}\\A_i\\...\\A_j\\...\end{pmatrix} = det\begin{pmatrix}...\\A_i\\A_{i + 1}\\...\\A_j\\...\end{pmatrix} = (-1)det\begin{pmatrix}...\\A_{i + 1}\\A_i\\...\\A_j\\...\end{pmatrix} =</math>

:::::::<math>(-1)^{j - i}det\begin{pmatrix}...\\A_{i + 1}\\...\\A_j\\A_i\\...\end{pmatrix} = (-1)^{j - i}(-1)^{j - i - 1}det\begin{pmatrix}...\\A_j\\A_{i + 1}\\...\\A_i\\...\end{pmatrix} =</math>

:::::::<math>(-1)^{2(j - i) - 1}det\begin{pmatrix}...\\A_j\\A_{i + 1}\\...\\A_i\\...\end{pmatrix} = (-1)^{- 1}det\begin{pmatrix}...\\A_j\\A_{i + 1}\\...\\A_i\\...\end{pmatrix} = (-1)det\begin{pmatrix}...\\A_j\\A_{i + 1}\\...\\A_i\\...\end{pmatrix} =</math>

:::::::<math>-det(B)</math>.

Then the proof of the theorem is complete.

==Nikita==
==Scanned Tutorial Notes by [[User Boyang.wu|Boyang.wu]]==
[[File:Tut.pdf]]

14-240/Tutorial-December 2

2014-12-08T18:17:01Z

Boyang.wu: /* Nikita */

{{14-240/Navigation}}

==Boris==

====Theorem====

Let <math>A</math> be a <math>n \times n</math> matrix and <math>B</math> be the matrix <math>A</math> with two rows interchanged. Then <math>det(A) = -det(B)</math>. Boris decided to prove the following lemma first:

=====Lemma 1=====

Let <math>A</math> be a <math>n \times n</math> matrix and <math>B</math> be the matrix <math>A</math> with two '''adjacent''' rows interchanged. Then <math>det(A) = -det(B)</math>.

All we need to show is that <math>det(A) + det(B) = 0</math>. Assume that <math>B</math> is the matrix <math>A</math> with rows <math>i, i + 1</math> of <math>A</math> interchanged. Since the determinant of a matrix with two identical rows is <math>0</math>, then:

:::::::<math>det(A) + det(B) =</math>

:::::::<math>det\begin{pmatrix}...\\A_i\\A_{i + 1}\\...\end{pmatrix} + det\begin{pmatrix}...\\A_{i + 1}\\A_i\\...\end{pmatrix} =</math>

:::::::<math>det\begin{pmatrix}...\\A_i\\A_i\\...\end{pmatrix} + det\begin{pmatrix}...\\A_i\\A_{i + 1}\\...\end{pmatrix} + det\begin{pmatrix}...\\A_{i + 1}\\A_{i + 1}\\...\end{pmatrix} + det\begin{pmatrix}...\\A_{i + 1}\\A_i\\...\end{pmatrix}</math>.

Since the determinant is linear in each row, then we continue where we left off:

:::::::<math>det(A) + det(B) =</math>

:::::::<math>det\begin{pmatrix}...\\A_i\\A_i\\...\end{pmatrix} + det\begin{pmatrix}...\\A_i\\A_{i + 1}\\...\end{pmatrix} + det\begin{pmatrix}...\\A_{i + 1}\\A_{i + 1}\\...\end{pmatrix} + det\begin{pmatrix}...\\A_{i + 1}\\A_i\\...\end{pmatrix} = </math>

:::::::<math>det\begin{pmatrix}...\\A_i\\A_i + A_{i + 1}\\...\end{pmatrix} + det\begin{pmatrix}...\\A_{i + 1}\\A_i + A_{i + 1}\\...\end{pmatrix} = det\begin{pmatrix}...\\A_i + A_{i + 1}\\A_i + A_{i + 1}\\...\end{pmatrix} = 0</math>.

Then <math>det(A) + det(B) = 0</math> and <math>det(A) = -det(B)</math>. The proof of the lemma is complete.

For the proof of the theorem, assume that <math>B</math> is the matrix <math>A</math> with rows <math>i, j</math> of <math>A</math> interchanged and <math>i \neq j</math>. By '''Lemma 1''', we have the following:

:::::::<math>det(A) =</math>

:::::::<math>(-1)det\begin{pmatrix}...\\A_{i + 1}\\A_i\\...\\A_j\\...\end{pmatrix} = det\begin{pmatrix}...\\A_i\\A_{i + 1}\\...\\A_j\\...\end{pmatrix} = (-1)det\begin{pmatrix}...\\A_{i + 1}\\A_i\\...\\A_j\\...\end{pmatrix} =</math>

:::::::<math>(-1)^{j - i}det\begin{pmatrix}...\\A_{i + 1}\\...\\A_j\\A_i\\...\end{pmatrix} = (-1)^{j - i}(-1)^{j - i - 1}det\begin{pmatrix}...\\A_j\\A_{i + 1}\\...\\A_i\\...\end{pmatrix} =</math>

:::::::<math>(-1)^{2(j - i) - 1}det\begin{pmatrix}...\\A_j\\A_{i + 1}\\...\\A_i\\...\end{pmatrix} = (-1)^{- 1}det\begin{pmatrix}...\\A_j\\A_{i + 1}\\...\\A_i\\...\end{pmatrix} = (-1)det\begin{pmatrix}...\\A_j\\A_{i + 1}\\...\\A_i\\...\end{pmatrix} =</math>

:::::::<math>-det(B)</math>.

Then the proof of the theorem is complete.

==Nikita==
==Scanned Lecture Notes by [[User Boyang.wu|Boyang.wu]]==
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File:Tut.pdf

2014-12-08T18:16:00Z

Boyang.wu:

14-240/Classnotes for Wednesday December 3

2014-12-08T17:24:02Z

Boyang.wu:

{{14-240/Navigation}}

Today's handout is {{Pensieve link|Classes/14-240/nb/Fibonacci.pdf|Fibonacci.pdf}}.

{{14-240:Dror/Students Divider}}
==Scanned Lecture Notes by [[User Boyang.wu|Boyang.wu]]==
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14-240/Classnotes for Monday November 10

2014-12-08T17:23:24Z

Boyang.wu:

The pdf notes for Wednesday class is: {{14-240/Navigation}}

==Scanned Lecture Notes by [[User Boyang.wu|Boyang.wu]]==
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By Yue Jiang--[[User:Yue.Jiang|Yue.Jiang]] ([[User talk:Yue.Jiang|talk]]) 11:29, 22 October 2014 (EDT)

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File:monday Nov 10.jpg|Nov 10 note

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