Drorbn - User contributions [en]

06-240/Final Exam Preparation Forum

2006-12-11T23:08:17Z

70.50.137.35: quick reply

{{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 (solved/unsolved, as judge by whoever created the question), 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.

===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}\\ia_{12}&a_{22}&ia_{22}+a_{32}\end{pmatrix}</math>

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

===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 and U is invertible, therefore A and T is similar, thus A is diagonalizable. Please comment. Thanks.

===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.

===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.

==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.

06-240/Class Photo

2006-12-11T20:50:58Z

70.50.137.35: (minor) quick reordering of names :/ (again)

Our class on September 28, 2006:

[[Image:06-240-ClassPhoto.jpg|thumb|centre|500px|Class Photo: click to enlarge]]
{{06-240/Navigation}}

Please identify yourself in this photo! There are two ways to do that:

* [[Special:Userlogin|Log in]] to this Wiki and edit this page. Put your name, userid, email address and location in the picture in the alphabetical list below.
* Send [[User:Drorbn|Dror]] an email message with this information.

The first option is more fun but less private.

===Who We Are...===

{| align=center border=1
|-
!First name
!Last name
!UserID
!Email
!In the photo
!Comments
{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn@ math.toronto.edu|location=facing everybody, as the photographer|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank}}
{{Photo Entry|last=Beach|first=Laura|userid=Beacher|email=laura.beach@ utoronto.ca|location=3rd row from front, far right in light shirt, in front of guy in hat.|comments=Lookin' good, everybody!}}
{{Photo Entry|last=Bjorndahl|first=Paolo|userid=Bjorndahl|email=paolo.bjorndahl@ utoronto.ca|location=Back row, 5th form the far right.|comments=}}
{{Photo Entry|last=Bumagin|first=Mark|userid=Bumaginm|email=mark.bumagin@ utoronto.ca|location=Guy in light blue collar shirt, second row, third on the right|comments= }}
{{Photo Entry|last=Cao|first=Yanshuai|userid=yanshuaicao|email=yanshuaicao@gmail.com|location= the Asian guy in jeans and jacket in the left mini photo,second from the left, |comments= Glad to be in this group }}
{{Photo Entry|last=Carberry|first=Mick|userid=MC|email=Mick.Carberry@ utoronto.ca|location=long haired, bearded old guy in back|comments= }}
{{Photo Entry|last=Cerezo|first=Richard|userid=Cerezo|email=richard.cerezo@ utoronto.ca|location=Guy in black jacket and black hat on far right, second from the bottom. |comments= }}
{{Photo Entry|last=Dong|first=Qi (Tom)|userid=houseofglass|email=tom_dongqi@ hotmail.com|location= The only chinese blonde guy with glasses located on the second row in the main picture|comments=testing 1 2 3 4 5 6 7 8 9 }}
{{Photo Entry|last=Dzamba|first=Michael|userid=dzambami|email=michael.dzamba@ utoronto.ca|location=In the middle, sort of, have a bit of extra hair on my head, forms sort of a spherical volume, sometimes visible from a distance|comments="The obvious mathematical breakthrough would be development of an easy way to factor large prime numbers." - Bill Gates , The Road Ahead (his book), P265, This is not a typo on my behalf ;) }}
{{Photo Entry|last=Gokmen|first=Murat|userid=Gokmen|email=uoftmurat@ gmail.com|location=the guy second row @second from the right:with the shining jacket & blue hat|comments=}}
{{Photo Entry|last=Guillon|first=Adrien|userid=adrien_joel|email=guillona_at_remove_me_utoronto.ca|location=Third row of main picture, third from the left with black jacket and brown hair.|comments= }}
{{Photo Entry|last=Halacheva|first=Iva|userid=Haliv|email=iva.halacheva@ utoronto.ca|location=Third row from the front, light brown jacket|comments= }}
{{Photo Entry|last=Harbans|first=Brad|userid=harbansb|email=harbansb@msn.com |location=not in picture|comments= Not in picture, but want good deed points credited}}
{{Photo Entry|last=Hoang|first=Uyen|userid=uhoang36|email=uhoang36@yahoo.com |location=The Asian girl in the front row.|comments= }}
{{Photo Entry|last=Jaimungal|first=Curt|userid=curtdbz|email=curtdbz@hotmail.com or curt.jaimungal@utoronto.ca|location=The brown guy in the middle of the mini-pic (bottom left) in the middle, without glasses.|comments= Hello.. Newman..!}}
{{Photo Entry|last=Kaifosh|first=Patrick|userid=Pat|email=patrick.kaifosh@ utoronto.ca|location=Front row, rightmost.|comments= }}
{{Photo Entry|last=Kim|first=Taehyung|userid=Taehyung Kim|email=taehyung.Kim@utoronto.ca|location= Asian,Second row from the back, Second right most, black shirt,black backpack|comments= }}
{{Photo Entry|last=Koziar|first=John|userid=John.koziar|email=|location=Front row, centre.|comments= }}
{{Photo Entry|last=Matskin|first=Jeffrey|userid=jeff.matskin|email=jeff.matskin@utoronto.ca|location= Bottom right corner of people, farthest to the right, white shirt|comments= }}
{{Photo Entry|last=McIntyre|first=Sean|userid=Smcintyre|email=s.mcintyre@ utoronto.ca|location=mini-picture, fourth from the right|comments= }}
{{Photo Entry|last=Merchant|first=Aliya|userid=amerchant|email=aliya.merchant@gmail.com|location=Main picture, third girl in the third row. The tiny girl who looks like she's hiding.|comments= }}
{{Photo Entry|last=Ng|first=Gilbert|userid=Gilbert|email=gilbert.ng@ utoronto.ca|location=to the right of the pole in the back, last row|comments= }}
{{Photo Entry|last=Nikolaev|first=Nikita|userid=Zimba|email=nnabelka@yahoo.ca|location=Second row, second from the right. A guy in bright-red and dark-blue jacket|comments=Thou shall love squirrels}}
{{Photo Entry|last=Park|first=Philip|userid=Psp|email=philip.park@utoronto.ca|location=2nd row from the back, 3rd from the right, wearing a light-blue shirt with a dark-blue jacket|comments=}}
{{Photo Entry|last=Qiao|first=Li|userid=joy9999|email=li.qiao@ utoronto.ca|location=behind the guy in black, last row|comments= }}
{{Photo Entry|last=Shestakov|first=Ilya|userid=ilya87|email=ilya.shestakov@utoronto.ca|location=In the yellow/green shirt in the lower right corner|comments= }}
{{Photo Entry|last=Soreanu|first=Alla|userid=Alla|email=alla.soreanu@ utoronto.ca|location=mini-picture, first from the left|comments= }}
{{Photo Entry|last=SUN|first=LUYANG|userid=Luyang|email=luyang.sun@ utoronto.ca|location=the guy in the middle, wearing a white coat row3 col4|comments=Hello World}}
{{Photo Entry|last=Tai|first=Johnathan|userid=Zapyre|email=zapyre_1@ hotmail.com|location= At the very back, left hand corner (all blue) |comments= >.<}}
{{Photo Entry|last=Veytsman|first=Maxim|userid=Mveytsman|email=|location=small picture in the bottom left corner. Second one from the right.|comments=}}
{{Photo Entry|last=Wan|first=Mike|userid=wanmike|email=Mike.Wan@utoronto.ca|location=Main picture, front row, second from right.|comments="If I have seen farther, it is by standing on the shoulders of giants." -Sir I N.}}
{{Photo Entry|last=Wong|first=Pak|userid=wongpak|email=pl.wong@ utoronto.ca|location=Third row from the back, left most, black shirt|comments= }}

06-240/Final Exam Preparation Forum

2006-12-11T20:48:11Z

70.50.137.35: it was ment to be #7 :/ (oops)

06-240/Final Exam Preparation Forum

2006-12-11T20:37:39Z

70.50.137.35: last minor change :/

06-240/Final Exam Preparation Forum

2006-12-11T20:36:53Z

70.50.137.35:

06-240/Final Exam Preparation Forum

2006-12-11T20:35:30Z

70.50.137.35:

06-240/Final Exam Preparation Forum

2006-12-11T20:34:58Z

70.50.137.35: minor changes to question to make it more readable

06-240/Final Exam Preparation Forum

2006-12-11T20:34:15Z

70.50.137.35:

06-240/Final Exam Preparation Forum

2006-12-11T20:33:15Z

70.50.137.35: Exam April/May 2006 #8 question