06-240/Classnotes For Thursday November 9

2006-11-10T04:40:07Z

Psp: 'Multiply a row/column by a nonzero scalar'

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==Review of Last Class==
{|
|-
|'''Problem.''' Find the rank (the dimension of the image) of a linear transformation <math>T</math> whose matrix representation is the matrix A shown on the right.
|<math>A=\begin{pmatrix}0&2&4&2&2\\4&4&4&8&0\\8&2&0&10&2\\6&3&2&9&1\end{pmatrix}</math>.
|}

{|
|- valign=bottom
|'''Theorem 1.''' If <math>T:V\to W</math> is a linear transformation and <math>P:V\to V</math> and <math>Q:W\to W</math> are ''invertible'' linear transformations, then the rank of <math>T</math> is the same as the rank of <math>QTP</math>.
|   
|'''Proof.''' Owed.
|- valign=bottom
|'''Theorem 2.''' The following row/column operations can be applied to a matrix <math>A</math> by multiplying it on the left/right (respectively) by certain ''invertible'' "elementary matrices":
# Swap two rows/columns
# Multiply a row/column by a nonzero scalar.
# Add a multiple of one row/column to another row/column.
|   
|'''Proof.''' Semi-owed.
|}

'''Solution of the problem.''' using these (invertible!) row/column operations we aim to bring <math>A</math> to look as close as possible to an identity matrix, hoping it will be easy to determine the rank of the matrix we get at the end:

{| border="1px" cellspadding="5" cellspacing=0 style="font-size:90%;"
|+
|align=center|'''Do'''
|align=center|'''Get'''
|align=center|'''Do'''
|align=center|'''Get'''
|- valign=top
|1. Bring a <math>1</math> to the upper left corner by swapping the first two rows and multiplying the first row (after the swap) by <math>1/4</math>.
|align=center|<math>\begin{pmatrix}1&1&1&2&0\\0&2&4&2&2\\8&2&0&10&2\\6&3&2&9&1\end{pmatrix}</math>
|2. Add <math>(-8)</math> times the first row to the third row, in order to cancel the <math>8</math> in position 3-1.
|align=center|<math>\begin{pmatrix}1&1&1&2&0\\0&2&4&2&2\\0&-6&-8&-6&2\\6&3&2&9&1\end{pmatrix}</math>
|- valign=top
|3. Likewise add <math>(-6)</math> times the first row to the fourth row, in order to cancel the <math>6</math> in position 4-1.
|align=center|<math>\begin{pmatrix}1&1&1&2&0\\0&2&4&2&2\\0&-6&-8&-6&2\\0&-3&-4&-3&1\end{pmatrix}</math>
|4. With similar column operations (you need three of those) cancel all the entries in the first row (except, of course, the first, which is used in the canceling).
|align=center|<math>\begin{pmatrix}1&0&0&0&0\\0&2&4&2&2\\0&-6&-8&-6&2\\0&-3&-4&-3&1\end{pmatrix}</math>
|- valign=top
|5. Turn the 2-2 entry to a <math>1</math> by multiplying the second row by <math>1/2</math>.
|align=center|<math>\begin{pmatrix}1&0&0&0&0\\0&1&2&1&1\\0&-6&-8&-6&2\\0&-3&-4&-3&1\end{pmatrix}</math>
|6. Using two row operations "clean" the second column; that is, cancel all entries in it other than the "pivot" <math>1</math> at position 2-2.
|align=center|<math>\begin{pmatrix}1&0&0&0&0\\0&1&2&1&1\\0&0&4&0&8\\0&0&2&0&4\end{pmatrix}</math>
|- valign=top
|7. Using three column operations clean the second row except the pivot.
|align=center|<math>\begin{pmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&4&0&8\\0&0&2&0&4\end{pmatrix}</math>
|8. Clean up the row and the column of the <math>4</math> in position 3-3 by first multiplying the third row by <math>1/4</math> and then performing the appropriate row and column transformations. Notice that by pure luck, the <math>4</math> at position 4-5 of the matrix gets killed in action.
|align=center|<math>\begin{pmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&0&0\end{pmatrix}</math>
|}

But the matrix we now have represents a linear transformation <math>S</math> satisfying <math>S(v_1,\,v_2,\,v_3,\,v_4\,v_5)=(w_1,\,w_2,\,w_3,\,0,\,0)</math> for some bases <math>(v_i)_{i=1}^5</math> of <math>V</math> and <math>(w_j)_{j=1}^4</math> of <math>W</math>. Thus the image (range) of <math>S</math> is spanned by <math>\{w_1,w_2,w_3\}</math>, and as these are independent, they form a basis of the image. Thus the rank of <math>S</math> is <math>3</math>. Going backward through the "matrix reduction" process above and repeatedly using theorems 1 and 2, we find that the rank of <math>T</math> must also be <math>3</math>.

06-240/Class Photo

2006-10-07T20:55:06Z

Psp: Added Philip

Our class on September 28, 2006:

[[Image:06-240-ClassPhoto.jpg|thumb|centre|500px|Class Photo: click to enlarge]]
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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=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=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=Halacheva|first=Iva|userid=Haliv|email=iva.halacheva@ utoronto.ca|location=Third row from the front, light brown jacket|comments= }}
{{Photo Entry|last=Kaifosh|first=Patrick|userid=Pat|email=patrick.kaifosh@ utoronto.ca|location=Front row, rightmost.|comments= }}
{{Photo Entry|last=Koziar|first=John|userid=John.koziar|email=|location=Front row, centre.|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=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=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=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=Wong|first=Pak|userid=wongpak|email=plwong@ utoronto.ca|location=Third row from the back, left most, black shirt|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=Matskin|first=Jeffrey|userid=jeff.matskin|email=jeff.matskin@utoronto.ca|location= Bottom right corner of people, farthest to the right, white shirt|comments= }}

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06-240/Classnotes For Thursday November 9

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