Drorbn - User contributions [en]

File:MAT Tut012.pdf

2006-12-02T19:55:40Z

Alla: Week 12 tutorial notes

Week 12 tutorial notes

File:MAT Lect021.pdf

2006-12-02T19:51:05Z

Alla: Week 12 lecture 2 notes

Week 12 lecture 2 notes

File:MAT Lect020.pdf

2006-12-02T19:50:19Z

Alla: Week 12 lecture 1 notes

Week 12 lecture 1 notes

06-240/Classnotes For Tuesday November 28

2006-12-02T19:49:19Z

Alla: corrected file name

{{06-240/Navigation}}

==Class notes==

[[Media:MAT Lect020.pdf|Scan of Week 12 Lecture 1 notes]]

06-240/Classnotes For Tuesday November 28

2006-12-02T19:48:56Z

Alla: added navigation panel

{{06-240/Navigation}}

==Class notes==

[[Media:MAT Lect018.pdf|Scan of Week 11 Lecture 1 notes]]

06-240/Classnotes For Thursday November 30

2006-12-02T19:48:08Z

Alla: Added scan of Week 12 lecture and tutorial notes

{{06-240/Navigation}}

==Class Notes==

[[Media:MAT Lect021.pdf|Scan of Week 12 Lecture 2 notes]]

==Tutorial Notes==

[[Media:MAT Tut012.pdf|Scan of Week 12 Tutorial notes]]

06-240/Classnotes For Tuesday November 28

2006-12-02T19:46:46Z

Alla: Added scan of Week 12 lecture notes

==Class notes==

[[Media:MAT Lect018.pdf|Scan of Week 11 Lecture 1 notes]]

Template:06-240/Navigation

2006-12-02T19:44:44Z

Alla: Added days to panel

{| 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>
<div class="NavContent">
{| border="1px" cellpadding="1" cellspacing="0" width="220" style="margin: 0 0 1em 0.5em; font-size: small"
|-
!#
!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
|
|-
|align=center|F
|Dec 11
|Final: Dec 13 2-5PM at BN3
|-
|colspan=3 align=center|[[06-240/Register of Good Deeds|Register of Good Deeds]]
|-
|colspan=3 align=center|[[Image:06-240-ClassPhoto.jpg|180px]]<br>[[06-240/Class Photo|Add your name / see who's in!]]
|-
|colspan=3 align=center|[[Template:06-240/Navigation|edit the panel]]
|}
</div></div>
|}

06-240/Classnotes For Thursday October 19

2006-11-26T17:45:24Z

Alla: Added Navigation Panel

{{06-240/Navigation}}

[[Media:MAT Lect010.pdf|Scan of Week 6 Lecture 2 notes]]

[[Media:MAT Tut005.pdf|Scan of Week 6 Tutorial notes]]

06-240/Classnotes For Tuesday November 7

2006-11-26T17:44:53Z

Alla: Added Navigation Panel

{{06-240/Navigation}}

[Nov 7th Lecture Notes[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:Scan0001.jpg]]
[Nov 7th Lecture Notes[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:Scan0002.jpg]]
[Nov 7th Lecture Notes[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:Scan0003.jpg]]
[Nov 7th Lecture Notes[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:Scan0004.jpg]]

[[Media:Lect 014.pdf|Scan of Week 9 Lecture 1 notes]]

06-240/Classnotes For Thursday November 23

2006-11-26T17:44:20Z

Alla: Added Navigation Panel

{{06-240/Navigation}}

==Class Notes==

[[Media:MAT Lect019.pdf|Scan of Week 11 Lecture 2 notes]]

==Tutorial Notes==

[[Media:MAT Tut011.pdf|Scan of Week 11 Tutorial notes]]

06-240/Classnotes For Thursday November 23

2006-11-26T17:42:48Z

Alla: Added scan of Week 11 Lecture and tutorial notes

==Class Notes==

[[Media:MAT Lect019.pdf|Scan of Week 11 Lecture 2 notes]]

==Tutorial Notes==

[[Media:MAT Tut011.pdf|Scan of Week 11 Tutorial notes]]

Template:06-240/Navigation

2006-11-26T17:39:56Z

Alla: Appropriate header

{| 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>
<div class="NavContent">
{| border="1px" cellpadding="1" cellspacing="0" width="220" style="margin: 0 0 1em 0.5em; font-size: small"
|-
!#
!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
|HW10
|-
|align=center|13
|Dec 4
|
|-
|align=center|F
|Dec 11
|Final: Dec 13 2-5PM at BN3
|-
|colspan=3 align=center|[[06-240/Register of Good Deeds|Register of Good Deeds]]
|-
|colspan=3 align=center|[[Image:06-240-ClassPhoto.jpg|180px]]<br>[[06-240/Class Photo|Add your name / see who's in!]]
|-
|colspan=3 align=center|[[Template:06-240/Navigation|edit the panel]]
|}
</div></div>
|}

File:MAT Tut011.pdf

2006-11-26T17:38:48Z

Alla: Week 11 Tutorial notes

Week 11 Tutorial notes

File:MAT Lect019.pdf

2006-11-26T17:36:48Z

Alla: Week 11 Lecture 2 notes

Week 11 Lecture 2 notes

Template:06-240/Navigation

2006-11-26T17:33:34Z

Alla: Added day to panel

{| 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>
<div class="NavContent">
{| border="1px" cellpadding="1" cellspacing="0" width="220" style="margin: 0 0 1em 0.5em; font-size: small"
|-
!#
!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/|Thu]]
|-
|align=center|12
|Nov 27
|HW10
|-
|align=center|13
|Dec 4
|
|-
|align=center|F
|Dec 11
|Final: Dec 13 2-5PM at BN3
|-
|colspan=3 align=center|[[06-240/Register of Good Deeds|Register of Good Deeds]]
|-
|colspan=3 align=center|[[Image:06-240-ClassPhoto.jpg|180px]]<br>[[06-240/Class Photo|Add your name / see who's in!]]
|-
|colspan=3 align=center|[[Template:06-240/Navigation|edit the panel]]
|}
</div></div>
|}

File:MAT Lect018.pdf

2006-11-26T17:28:16Z

Alla: Week 11 Lecture 1 notes

Week 11 Lecture 1 notes

06-240/Classnotes For Tuesday November 21

2006-11-26T17:25:50Z

Alla: Uploaded Week 11 Lecture 1 notes

{{06-240/Navigation}}

==More about the [[User:Wongpak|Wongpak]] Matrices==

In [[Talk:06-240/Classnotes_For_Tuesday_November_14]], [[User:Wongpak]] asked something about row echelon form and reduced row echelon form, and gave the following matrices as specific examples:

{| align=center
|-
|<math>A_1=\begin{pmatrix}1&3&2&4&2\\0&1&2&3&4\\0&0&0&1&2\\0&0&0&0&0 \end{pmatrix}</math>
|width=10%|
|<math>A_2=\begin{pmatrix}1&0&-4&0&-6\\0&1&2&0&-2\\0&0&0&1&2\\0&0&0&0&0 \end{pmatrix}</math>
|}

So let us assume row reduction leads us to the systems <math>A_1x=b</math> or <math>A_2x=b</math>. What does it tell us about the solutions? Let us start from the second system:

{| align=center
|- align=center
|<math>\begin{pmatrix}1&0&-4&0&-6\\0&1&2&0&-2\\0&0&0&1&2\\0&0&0&0&0\end{pmatrix} \begin{pmatrix}x_1\\x_2\\x_3\\x_4\\x_5\end{pmatrix} = \begin{pmatrix}b_1\\b_2\\b_3\\b_4\end{pmatrix}</math>
|width=15%|or
|
{|
|-
|<math>x_1</math>
|
|<math>-4x_3</math>
|
|<math>-6x_5</math>
|<math>=</math>
|<math>b_1</math>
|-
|
|<math>x_2</math>
|<math>+2x_3</math>
|
|<math>-2x_5</math>
|<math>=</math>
|<math>b_2</math>
|-
|
|
|
|<math>x_4</math>
|<math>+2x_5</math>
|<math>=</math>
|<math>b_3</math>
|-
|
|
|
|
|align=center|<math>0</math>
|<math>=</math>
|<math>b_4</math>
|}
|}

Well, quite clearly if <math>b_4\neq 0</math> this system has no solutions, but if <math>b_4=0</math> it has solutions no matter what <math>b_1</math>, <math>b_2</math> and <math>b_3</math> are. Finally, for any given values of <math>b_1</math>, <math>b_2</math> and <math>b_3</math> we can choose the values of <math>x_3</math> and <math>x_5</math> (the variables corresponding the columns containing no pivots) as we please, and then get solutions by setting the "pivotal variables" in terms of the non-pivotal ones as follows: <math>x_1=b_1+4x_3+6x_5</math>, <math>x_2=b_2-2x_3+2x_5</math> and <math>x_4=b_3-2x_5</math>.

What about the system corresponding to <math>A_1</math>? It is
{| align=center
|- align=center
|<math>\begin{pmatrix}1&3&2&4&2\\0&1&2&3&4\\0&0&0&1&2\\0&0&0&0&0\end{pmatrix} \begin{pmatrix}x_1\\x_2\\x_3\\x_4\\x_5\end{pmatrix} = \begin{pmatrix}b_1\\b_2\\b_3\\b_4\end{pmatrix}</math>
|width=15%|or
|
{|
|- align=center
|<math>x_1</math>
|<math>+3x_2</math>
|<math>+2x_3</math>
|<math>+4x_4</math>
|<math>+2x_5</math>
|<math>=</math>
|<math>b_1</math>
|- align=center
|
|<math>x_2</math>
|<math>+2x_3</math>
|<math>+3x_4</math>
|<math>+4x_5</math>
|<math>=</math>
|<math>b_2</math>
|- align=center
|
|
|
|<math>x_4</math>
|<math>+2x_5</math>
|<math>=</math>
|<math>b_3</math>
|-
|
|
|
|
|align=center|<math>0</math>
|<math>=</math>
|<math>b_4</math>
|}
|}

Here too we have solutions iff <math>b_4=0</math>, and if <math>b_4=0</math>, we have the freedom to choose the non-pivotal variables <math>x_3</math> and <math>x_5</math> as we please. But now the formulas for fixing the pivotal variables <math>x_1</math>, <math>x_2</math> and <math>x_4</math> in terms of the non-pivotal ones are a bit harder.

==Class notes==

[[Media:MAT Lect018.pdf|Scan of Week 11 Lecture 1 notes]]

06-240/Classnotes For Tuesday November 21

2006-11-26T17:24:15Z

Alla: added header

06-240/Classnotes For Thursday November 9

2006-11-26T17:23:05Z

Alla: formatted

{{06-240/Navigation}}

==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>.

==Class Notes==

[[Media:Lect015.pdf|Scan of Week 9 Lecture 2 notes]]

==Tutorial Notes==

[[Media:06-240-nov09tut-1.jpeg|Nov09 Lecture notes 1 of 3]]

[[Media:06-240-nov09tut-2.jpeg|Nov09 Lecture notes 2 of 3]]

[[Media:06-240-nov09tut-3.jpeg|Nov09 Lecture notes 3 of 3]]

[[Media:Tut009.pdf|Scan of Week 9 Tutorial notes]]

File:MAT Tut010.pdf

2006-11-26T17:18:40Z

Alla: Week 10 Tutorial

Week 10 Tutorial

File:MAT Lect017.pdf

2006-11-26T17:15:53Z

Alla: Week 10 Lecture 2 notes

Week 10 Lecture 2 notes

06-240/Classnotes For Thursday November 16

2006-11-26T17:15:04Z

Alla: Uploaded Week 10 Lecture 2 notes and Tutorial notes

{{06-240/Navigation}}

[[Media:06-240-nov16lec-1.jpeg|Nov16 Lecture notes: 1 of 2]]

[[Media:06-240-nov16lec-2.jpeg|Nov16 Lecture notes: 2 of 2]]

[[Media:06-240-nov16tut-1.jpeg|Nov16 Tutorial notes: 1 of 4]]

[[Media:06-240-nov16tut-2.jpeg|Nov16 Tutorial notes: 2 of 4]]

[[Media:06-240-nov16tut-3.jpeg|Nov16 Tutorial notes: 3 of 4]]

[[Media:06-240-nov16tut-4.jpeg|Nov16 Tutorial notes: 4 of 4]]

[[Media:MAT Lect017.pdf|Scan of Week 10 Lecture 2 notes]]

[[Media:MAT Tut010.pdf|Scan of Week 10 Tutorial notes]]

06-240/Classnotes For Tuesday November 14

2006-11-26T17:12:09Z

Alla: formatted

{{06-240/Navigation}}

[[Media:06-240-nov14lec-1.jpeg|Nov14 Lecture notes: 1 of 4]]

[[Media:06-240-nov14lec-2.jpeg|Nov14 Lecture notes: 2 of 4]]

[[Media:06-240-nov14lec-3.jpeg|Nov14 Lecture notes: 3 of 4]]

[[Media:06-240-nov14lec-4.jpeg|Nov14 Lecture notes: 4 of 4]]

[[Media:06-240-november14th-lecture.pdf|November14th-Lecture notes]]

[[Media:MAT Lect016.pdf|Scan of Week 10 Lecture 1 notes]]

File:MAT Lect016.pdf

2006-11-26T17:10:51Z

Alla: Week 10 Lecture 1 notes

Week 10 Lecture 1 notes

06-240/Classnotes For Tuesday November 14

2006-11-26T17:10:13Z

Alla: Uploaded Week 10 Lecture 1 notes

06-240/Classnotes For Tuesday November 7

2006-11-15T03:52:28Z

Alla:

[Nov 7th Lecture Notes[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:Scan0001.jpg]]
[Nov 7th Lecture Notes[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:Scan0002.jpg]]
[Nov 7th Lecture Notes[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:Scan0003.jpg]]
[Nov 7th Lecture Notes[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:Scan0004.jpg]]

[[Media:Lect 014.pdf|Scan of Week 9 Lecture 1 notes]]

06-240/Classnotes For Tuesday November 7

2006-11-15T03:52:12Z

Alla: Uploaded Week 9 Lecture 1 notes

[Nov 7th Lecture Notes[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:Scan0001.jpg]]
[Nov 7th Lecture Notes[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:Scan0002.jpg]]
[Nov 7th Lecture Notes[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:Scan0003.jpg]]
[Nov 7th Lecture Notes[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:Scan0004.jpg]]
[[Media:Lect 014.pdf|Scan of Week 9 Lecture 1 notes]]

06-240/Classnotes For Thursday November 9

2006-11-15T03:50:50Z

Alla: Uploaded Week 9 Lecture 2 notes and Tutorial notes

{{06-240/Navigation}}

==Tutorial Notes==

[[Media:06-240-nov09tut-1.jpeg|Nov09 Lecture notes 1 of 3]]

[[Media:06-240-nov09tut-2.jpeg|Nov09 Lecture notes 2 of 3]]

[[Media:06-240-nov09tut-3.jpeg|Nov09 Lecture notes 3 of 3]]

[[Media:Lect015.pdf|Scan of Week 9 Lecture 2 notes]]

[[Media:Tut009.pdf|Scan of Week 9 Tutorial notes]]

==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>.

File:Lect015.pdf

2006-11-15T03:48:17Z

Alla: Week 9 Lect 2

Week 9 Lect 2

File:Lect 014.pdf

2006-11-15T03:47:43Z

Alla: Week 9 Lecture 1

Week 9 Lecture 1

File:Tut009.pdf

2006-11-15T03:46:59Z

Alla: Tutorial 9

Tutorial 9

File:MAT Lect013.pdf

2006-11-05T20:18:46Z

Alla:

06-240/Classnotes For Thursday November 2

2006-11-05T20:17:41Z

Alla: Added scan of Week 8 Lecture 2 notes

{{06-240/Navigation}}

[[Media:06-240-nov2lec.pdf|Nov02 Lecture notes]]

[[Media:06-240-02-November.pdf|November2nd-Lecture notes]]

[[Media:06-240-02-Nov.tut.pdf|November2nd-Tutorial notes]]

[[Media:MAT Lect013.pdf|Scan of Week 8 Lecture 2 notes]]

[[Media:MAT Tut008.pdf|Scan of Week 8 Tutorial notes]]

File:MAT Tut008.pdf

2006-11-05T20:15:01Z

Alla:

File:MAT Tut006.pdf

2006-10-26T23:20:14Z

Alla: Week 7 Tutorial notes

Week 7 Tutorial notes

File:MAT Lect011.pdf

2006-10-26T23:19:53Z

Alla: Week 7 Lecture notes

Week 7 Lecture notes

06-240/Classnotes For Thursday October 26

2006-10-26T23:11:17Z

Alla:

[[Media:MAT Lect011.pdf|Scan of Week 7 Lecture notes]]

[[Media:MAT Tut006.pdf|Scan of Week 7 Tutorial notes]]

Template:06-240/Navigation

2006-10-26T23:10:05Z

Alla:

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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]], Extra Hour
|-
|align=center|8
|Oct 30
|HW6
|-
|align=center|9
|Nov 6
|HW7
|-
|align=center|10
|Nov 13
|HW8
|-
|align=center|11
|Nov 20
|HW9
|-
|align=center|12
|Nov 27
|HW10
|-
|align=center|13
|Dec 4
|
|-
|align=center|F
|Dec 11
|Final: Dec 13 2-5PM at BN3
|-
|colspan=3 align=center|[[06-240/Register of Good Deeds|Register of Good Deeds]]
|-
|colspan=3 align=center|[[Image:06-240-ClassPhoto.jpg|180px]]<br>[[06-240/Class Photo|Add your name / see who's in!]]
|}
</div></div>
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06-240/Classnotes For Thursday October 19

2006-10-26T23:07:50Z

Alla: Added scan of Week 6 Tutorial notes

[[Media:MAT Lect010.pdf|Scan of Week 6 Lecture 2 notes]]

[[Media:MAT Tut005.pdf|Scan of Week 6 Tutorial notes]]

06-240/Classnotes For Thursday October 19

2006-10-26T23:03:55Z

Alla: Added Week 6 Lecture 2 ntoes

[[Media:MAT Lect010.pdf|Scan of Week 6 Lecture 2 notes]]

Template:06-240/Navigation

2006-10-26T23:00:45Z

Alla: Added Oct 19 link

{| cellpadding="0" cellspacing="0" style="clear: right; float: right"
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<div class="NavContent">
{| border="1px" cellpadding="1" cellspacing="0" width="220" style="margin: 0 0 1em 0.5em; font-size: small"
|-
!#
!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]], Extra Hour
|-
|align=center|8
|Oct 30
|HW6
|-
|align=center|9
|Nov 6
|HW7
|-
|align=center|10
|Nov 13
|HW8
|-
|align=center|11
|Nov 20
|HW9
|-
|align=center|12
|Nov 27
|HW10
|-
|align=center|13
|Dec 4
|
|-
|align=center|F
|Dec 11
|Final: Dec 13 2-5PM at BN3
|-
|colspan=3 align=center|[[06-240/Register of Good Deeds|Register of Good Deeds]]
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06-240/Classnotes For Tuesday October 10

2006-10-13T00:27:30Z

Alla: Change heading title

{{06-240/Navigation}}

====Scan of Lecture Notes====

* PDF file by [[User:Alla]]: [[Media:MAT_Lect009.pdf|Week 5 Lecture 1 notes]]

==A Quick Summary by {{Dror}}==
(Intentionally terse. A sea of details appears in the book and already appeared on the blackboard. But these are useless without some '''organizing principles'''; in some sense, "understanding" is precisely being able to see those principles within the sea of details. Yet don't fool yourself into thinking that the principles are enough even without the details!)

'''Theorem.''' A finite generating set <math>G</math> has a subset which is a basis.

'''Proof Sketch.''' Grab more and more elements of <math>G</math> so long as they are linearly independent. When you can't any more, you have a basis.

'''Lemma.''' (The Replacement Lemma) If <math>G</math> generates and <math>L</math> is linearly independent, then <math>|L|\leq|G|</math> and you can replace <math>|L|</math> of the elements of <math>G</math> by the elements of <math>L</math>, and still have a generating set.

'''Proof Sketch.''' Insert the elements of <math>L</math> one by one, and for each one that comes in, take one out of <math>G</math>. Which one? One that is used in expressing the newcomer in terms of the vector currently being inserted. Such one must exist or else the newcomer is a linear combination of some of the elements of <math>L</math>, but <math>L</math> is linearly independent.

'''Theorem.''' If a vector space <math>V</math> has a finite basis, all bases thereof are finite and have the same number of elements, the "dimension of <math>V</math>".

'''Proof Sketch.''' By replacement, <math>|\alpha|\leq|\beta|</math> and <math>|\beta|\leq|\alpha|</math>.

'''Theorem.''' Assume <math>\dim V=n</math>.
# If <math>G</math> generates, <math>|G|\geq n</math>. In case of equality, <math>G</math> is a basis.
# If <math>L</math> is linearly independent, <math>|L|\leq n</math>. In case of equality, <math>L</math> is a basis.

'''Proof Sketch.'''
# Find a basis within <math>G</math>; it has <math>n</math> elements.
# Use replacement to place the elements of <math>L</math> within some basis.

06-240/Classnotes For Tuesday October 10

2006-10-13T00:21:36Z

Alla: Posted Week 5 Lecture 1 notes

{{06-240/Navigation}}

===Links to Classnotes===
* PDF file by [[User:Alla]]: [[Media:MAT_Lect009.pdf|Week 5 Lecture 1 notes]]

==A Quick Summary by {{Dror}}==
(Intentionally terse. A sea of details appears in the book and already appeared on the blackboard. But these are useless without some '''organizing principles'''; in some sense, "understanding" is precisely being able to see those principles within the sea of details. Yet don't fool yourself into thinking that the principles are enough even without the details!)

'''Theorem.''' A finite generating set <math>G</math> has a subset which is a basis.

'''Proof Sketch.''' Grab more and more elements of <math>G</math> so long as they are linearly independent. When you can't any more, you have a basis.

'''Lemma.''' (The Replacement Lemma) If <math>G</math> generates and <math>L</math> is linearly independent, then <math>|L|\leq|G|</math> and you can replace <math>|L|</math> of the elements of <math>G</math> by the elements of <math>L</math>, and still have a generating set.

'''Proof Sketch.''' Insert the elements of <math>L</math> one by one, and for each one that comes in, take one out of <math>G</math>. Which one? One that is used in expressing the newcomer in terms of the vector currently being inserted. Such one must exist or else the newcomer is a linear combination of some of the elements of <math>L</math>, but <math>L</math> is linearly independent.

'''Theorem.''' If a vector space <math>V</math> has a finite basis, all bases thereof are finite and have the same number of elements, the "dimension of <math>V</math>".

'''Proof Sketch.''' By replacement, <math>|\alpha|\leq|\beta|</math> and <math>|\beta|\leq|\alpha|</math>.

'''Theorem.''' Assume <math>\dim V=n</math>.
# If <math>G</math> generates, <math>|G|\geq n</math>. In case of equality, <math>G</math> is a basis.
# If <math>L</math> is linearly independent, <math>|L|\leq n</math>. In case of equality, <math>L</math> is a basis.

'''Proof Sketch.'''
# Find a basis within <math>G</math>; it has <math>n</math> elements.
# Use replacement to place the elements of <math>L</math> within some basis.

File:MAT Tut005.pdf

2006-10-13T00:20:14Z

Alla: Week 5 Tutorial

Week 5 Tutorial

File:MAT Lect010.pdf

2006-10-13T00:19:56Z

Alla: Week 5 Lecture 2

Week 5 Lecture 2

File:MAT Lect009.pdf

2006-10-13T00:19:36Z

Alla: Week 5 Lecture 1

Week 5 Lecture 1

File:MAT Tut004.pdf

2006-10-08T21:57:49Z

Alla: Week 4 Tutorial Notes

Week 4 Tutorial Notes

06-240/Classnotes For Thursday October 5

2006-10-08T21:56:08Z

Alla: Added scan of Week 4 Tutorial notes

===Links to Classnotes===

* PDF file by [[User:Alla]]: [[Media:MAT_Lect008.pdf|Week 4 Lecture 2 notes]]

===Scan of Tutorial notes===

* PDF file by [[User:Alla]]: [[Media:MAT_Tut004.pdf|Week 4 Tutorial notes]]

----<math>\mbox{From last class}{}_{}^{}</math>

<math>M_1=\begin{pmatrix}1&0\\0&0\end{pmatrix},
M_2=\begin{pmatrix}0&1\\0&0\end{pmatrix},
M_3=\begin{pmatrix}0&0\\1&0\end{pmatrix},
M_4\begin{pmatrix}0&0\\0&1\end{pmatrix}</math>

<math>N_1=\begin{pmatrix}0&1\\1&1\end{pmatrix},
N_2=\begin{pmatrix}1&0\\1&1\end{pmatrix},
N_3=\begin{pmatrix}1&1\\0&1\end{pmatrix},
N_4\begin{pmatrix}1&1\\1&0\end{pmatrix}</math>

<math>\mbox{The }M_i\mbox{s generate }M_{2\times 2}</math>

<math>\mbox{Fact }T\subset\mbox{ span }S\Rightarrow \mbox{ span }T\subset\mbox{ span }S </math>

<math>S\subset V\mbox{ is linearly independent }\Leftrightarrow \mbox{ whenever }u_i\in S\mbox{ are distinct}</math>

<math>\sum a_iu_i=0\Rightarrow V_ia_i=0 \mbox{ waste not}</math>
<br>

<br>
<math>\mbox{Comments}{}_{}^{}</math>
#<math>\emptyset\subset V\mbox{ is linearly independent}</math>
#<math>\lbrace u\rbrace\mbox{ is linearly independent iff }u_{}^{}\neq 0</math>
#<math>\mbox{If }S_1^{}\subset S_2\subset V</math>
##<math>\mbox{If }S_1^{}\mbox{ is linearly dependent, so is }S_2</math>
##<math>\mbox{If }S_2^{}\mbox{ is linearly dependent, so is }S_1</math>
##<math>\mbox{If }S_1^{}\mbox{ generates }V\mbox{, so does }S_2</math>
##<math>\mbox{If }S_2^{}\mbox{ does not generate }V\mbox{ neither does }S_1</math>
#<math>\mbox{If }S_{}^{}\mbox{ is linearly independent in }V\mbox{ and }v\notin S\mbox{ then }S\cup\lbrace u\rbrace\mbox{ is linearly independent.}</math>
<br>

<br>
<math>\mbox{Proof}{}_{}^{}</math>

<math>\mbox{1.}\Leftarrow:\mbox{ start from second assertion and deduce first.}</math>

<math>\mbox{Assume }v_{}^{}\in \mbox{span }S</math>
<math>v=\sum a_iu_i\mbox{ where }u_i\in S, a_i\in F</math>

<math>\sum a_iu_i-1\cdot v=0\mbox{ this is a linear combination of elements in }S\cup v</math>
<math>\mbox{ in which not all coefficients are }0 \mbox{ and which add to }0_{}^{}.</math>
<math>\mbox{So }S\cup \lbrace v\rbrace\mbox{ is linearly dependent by definition}</math>
<br>
<math>\mbox{2.}:\Rightarrow\mbox{ Assume }S\cup \lbrace v\rbrace\mbox{ is linearly dependent }\Rightarrow\mbox{ a linear combination can be found, of the form:}</math>

<math>(*)\qquad\sum a_iu_i+bv=0\mbox{ where }u_i\in S\mbox{ and not all of the }a_i \mbox{ and }b \mbox{ are }0</math>

<math>\mbox{If }b=0\mbox{, then }\sum a_iu_i=0\mbox{ and not }a_i\mbox{s are }0 </math>
<math>{}_{}^{}\Rightarrow S \mbox{ is linearly dependent}</math>
<math>{}_{}^{}\mbox{but initial assumption was }S\mbox{ is linearly independent.}\Rightarrow \mbox{ contradiction so }b\neq0</math>
<math>\mbox{So divide by }b\mbox{: (*) becomes }\sum\frac{a_i}{b}u_i + v = 0\Rightarrow v=-\sum\frac{a_i}{b}u_i\Rightarrow v\in \mbox{ span }S</math>
<br>

<br>
<math>\mbox{Definition}{}_{}^{}</math>

<math>{}_{}^{}\mbox{A basis of a vector space }V\mbox{ is a subset }\beta\subset V</math>
<math>{}_{}^{}\mbox{such that}</math>
#<math>{}_{}^{}\beta\mbox{ generates }V\mbox{ or }V=\mbox{ span }\beta</math>
#<math>{}_{}^{}\beta\mbox{ is linearly independent.}</math>
<br>

<br>
<math>\mbox{Examples}{}_{}^{}</math>

<math>1. \beta=\emptyset{}_{}^{}\mbox{ is a basis of }\lbrace0\rbrace</math>

<math>2. {}_{}^{}V\mbox{ be }\mathbb{R}\mbox{ as a vector space over }\mathbb{R}</math>
<math>\qquad{}_{}^{}\beta=\lbrace5\rbrace\mbox{ and }\beta=\lbrace1\rbrace\mbox{ are bases.}</math>

<math>3.{}_{}^{}\mbox{ Let }V\mbox{ be }\mathbb{C}\mbox{ as a vector space over }\mathbb{R} \quad\beta=\lbrace1,i\rbrace</math>

:<math>\qquad{}_{}^{}\mbox{Check}</math>

:<math>\qquad{}_{}^{}\mbox{1. Every complex number is a linear combination of }\beta.</math>
::<math>Z=a+bi=a\cdot 1+b\cdot i\mbox{ with coefficients in }\mathbb{R}\mbox{ so }\lbrace1,i\rbrace\mbox{ generates}</math>

:<math>\qquad{}_{}^{}\mbox{2. Show }\beta=\lbrace1,i\rbrace\mbox{ are linearly independent. Assume }a\cdot 1+b\cdot i=0\mbox{ where }a,b\in\mathbb{R}</math>
::<math>{}_{}^{}\Rightarrow a+bi=0\Rightarrow a=0\mbox{ and } b=0</math>

<math>{}_{}^{}\mbox{4. }V\in\mathbb{R}^n=
\left\lbrace\begin{pmatrix}\vdots\end{pmatrix}y,\qquad
e_1=\begin{pmatrix}1\\0\\\vdots\\0\end{pmatrix},
e_2=\begin{pmatrix}0\\1\\\vdots\\0\end{pmatrix},\ldots,
e_n=\begin{pmatrix}0\\0\\\vdots\\1\end{pmatrix}\right\rbrace</math>

:<math>{}_{}^{}e_1\ldots e_n\mbox{ are a basis of }V</math>
::<math>{}_{}^{}\mbox{They span }\begin{pmatrix}a_1\\\vdots\\a_n\end{pmatrix}=\sum a_ie_i</math>
::<math>{}_{}^{}\mbox{They are linearly independent. }\sum a_ie_i=0\Rightarrow \sum a_ie_i=
\begin{pmatrix}a_1\\\vdots\\a_n\end{pmatrix}=0\Rightarrow a_i=0 \quad\forall i</math>

<math>{}_{}^{}\mbox{5. In }V=P_3(\mathbb{R}),\qquad \beta=\lbrace 1,x,x^2,x^3\rbrace</math>

<math>{}_{}^{}\mbox{6. In }V=P_1(\mathbb{R})=\lbrace ax+b\rbrace,\qquad \beta=\lbrace 1+x,1-x\rbrace\mbox{ is a basis}</math>
:<math>{}_{}^{}\mbox{1. Generate }</math>
::<math>u_1+u_2=2\Rightarrow \frac{1}{2}(u_1+u_2)=1\mbox{ so }1 \in\mbox{ span }S</math>
::<math>u_1-u_2=2x\Rightarrow \frac{1}{2}(u_1-u_2)=x\mbox{ so }x \in\mbox{ span }S</math>
::<math>{}_{}^{}\mbox{ so span}\lbrace 1,x\rbrace \subset\mbox{ span }\beta</math>
:<math>{}_{}^{}\mbox{2. Linearly independent. Assume }au_1+bu_2=0</math>
::<math>\Rightarrow a(1+x)+b(1-x)=0\Rightarrow a+b+(a-b)x=0</math>
::<math>{}_{}^{}\Rightarrow a+b=0\mbox{ and }a-b=0</math>
::<math>(a+b)+(a-b)\Rightarrow 2a=0\Rightarrow a=0</math>
::<math>(a+b)-(a-b)\Rightarrow 2b=0\Rightarrow b=0</math>
<br>

<br>
<math>\mbox{Theorem}{}_{}^{}</math>

<math>{}_{}^{}\mbox{A subset }\beta\mbox{ of a vectorspace }V \mbox{ is a basis iff every }v\in V\mbox{ can be expressed as}</math>
<math>{}_{}^{}\mbox{a linear combination of elements in }</math>
<math>{}_{}^{}\beta \mbox{ in exactly one way.}</math>
<br>

<br>
<math>\mbox{Proof}{}_{}^{}</math>

<math>{}_{}^{}\mbox{It is a combination of things we already know.}</math>
#<math>{}_{}^{}\beta\mbox{ generates}</math>
#<math>{}_{}^{}\beta\mbox{ is linearly independent}</math>

06-240/Classnotes For Thursday October 5

2006-10-08T21:52:34Z

Alla: Added scan of Week 4 Lecture 2 notes

===Links to Classnotes===
* PDF file by [[User:Alla]]: [[Media:MAT_Lect008.pdf|Week 4 Lecture 2 notes]]
----<math>\mbox{From last class}{}_{}^{}</math>

<math>M_1=\begin{pmatrix}1&0\\0&0\end{pmatrix},
M_2=\begin{pmatrix}0&1\\0&0\end{pmatrix},
M_3=\begin{pmatrix}0&0\\1&0\end{pmatrix},
M_4\begin{pmatrix}0&0\\0&1\end{pmatrix}</math>

<math>N_1=\begin{pmatrix}0&1\\1&1\end{pmatrix},
N_2=\begin{pmatrix}1&0\\1&1\end{pmatrix},
N_3=\begin{pmatrix}1&1\\0&1\end{pmatrix},
N_4\begin{pmatrix}1&1\\1&0\end{pmatrix}</math>

<math>\mbox{The }M_i\mbox{s generate }M_{2\times 2}</math>

<math>\mbox{Fact }T\subset\mbox{ span }S\Rightarrow \mbox{ span }T\subset\mbox{ span }S </math>

<math>S\subset V\mbox{ is linearly independent }\Leftrightarrow \mbox{ whenever }u_i\in S\mbox{ are distinct}</math>

<math>\sum a_iu_i=0\Rightarrow V_ia_i=0 \mbox{ waste not}</math>
<br>

<br>
<math>\mbox{Comments}{}_{}^{}</math>
#<math>\emptyset\subset V\mbox{ is linearly independent}</math>
#<math>\lbrace u\rbrace\mbox{ is linearly independent iff }u_{}^{}\neq 0</math>
#<math>\mbox{If }S_1^{}\subset S_2\subset V</math>
##<math>\mbox{If }S_1^{}\mbox{ is linearly dependent, so is }S_2</math>
##<math>\mbox{If }S_2^{}\mbox{ is linearly dependent, so is }S_1</math>
##<math>\mbox{If }S_1^{}\mbox{ generates }V\mbox{, so does }S_2</math>
##<math>\mbox{If }S_2^{}\mbox{ does not generate }V\mbox{ neither does }S_1</math>
#<math>\mbox{If }S_{}^{}\mbox{ is linearly independent in }V\mbox{ and }v\notin S\mbox{ then }S\cup\lbrace u\rbrace\mbox{ is linearly independent.}</math>
<br>

<br>
<math>\mbox{Proof}{}_{}^{}</math>

<math>\mbox{1.}\Leftarrow:\mbox{ start from second assertion and deduce first.}</math>

<math>\mbox{Assume }v_{}^{}\in \mbox{span }S</math>
<math>v=\sum a_iu_i\mbox{ where }u_i\in S, a_i\in F</math>

<math>\sum a_iu_i-1\cdot v=0\mbox{ this is a linear combination of elements in }S\cup v</math>
<math>\mbox{ in which not all coefficients are }0 \mbox{ and which add to }0_{}^{}.</math>
<math>\mbox{So }S\cup \lbrace v\rbrace\mbox{ is linearly dependent by definition}</math>
<br>
<math>\mbox{2.}:\Rightarrow\mbox{ Assume }S\cup \lbrace v\rbrace\mbox{ is linearly dependent }\Rightarrow\mbox{ a linear combination can be found, of the form:}</math>

<math>(*)\qquad\sum a_iu_i+bv=0\mbox{ where }u_i\in S\mbox{ and not all of the }a_i \mbox{ and }b \mbox{ are }0</math>

<math>\mbox{If }b=0\mbox{, then }\sum a_iu_i=0\mbox{ and not }a_i\mbox{s are }0 </math>
<math>{}_{}^{}\Rightarrow S \mbox{ is linearly dependent}</math>
<math>{}_{}^{}\mbox{but initial assumption was }S\mbox{ is linearly independent.}\Rightarrow \mbox{ contradiction so }b\neq0</math>
<math>\mbox{So divide by }b\mbox{: (*) becomes }\sum\frac{a_i}{b}u_i + v = 0\Rightarrow v=-\sum\frac{a_i}{b}u_i\Rightarrow v\in \mbox{ span }S</math>
<br>

<br>
<math>\mbox{Definition}{}_{}^{}</math>

<math>{}_{}^{}\mbox{A basis of a vector space }V\mbox{ is a subset }\beta\subset V</math>
<math>{}_{}^{}\mbox{such that}</math>
#<math>{}_{}^{}\beta\mbox{ generates }V\mbox{ or }V=\mbox{ span }\beta</math>
#<math>{}_{}^{}\beta\mbox{ is linearly independent.}</math>
<br>

<br>
<math>\mbox{Examples}{}_{}^{}</math>

<math>1. \beta=\emptyset{}_{}^{}\mbox{ is a basis of }\lbrace0\rbrace</math>

<math>2. {}_{}^{}V\mbox{ be }\mathbb{R}\mbox{ as a vector space over }\mathbb{R}</math>
<math>\qquad{}_{}^{}\beta=\lbrace5\rbrace\mbox{ and }\beta=\lbrace1\rbrace\mbox{ are bases.}</math>

<math>3.{}_{}^{}\mbox{ Let }V\mbox{ be }\mathbb{C}\mbox{ as a vector space over }\mathbb{R} \quad\beta=\lbrace1,i\rbrace</math>

:<math>\qquad{}_{}^{}\mbox{Check}</math>

:<math>\qquad{}_{}^{}\mbox{1. Every complex number is a linear combination of }\beta.</math>
::<math>Z=a+bi=a\cdot 1+b\cdot i\mbox{ with coefficients in }\mathbb{R}\mbox{ so }\lbrace1,i\rbrace\mbox{ generates}</math>

:<math>\qquad{}_{}^{}\mbox{2. Show }\beta=\lbrace1,i\rbrace\mbox{ are linearly independent. Assume }a\cdot 1+b\cdot i=0\mbox{ where }a,b\in\mathbb{R}</math>
::<math>{}_{}^{}\Rightarrow a+bi=0\Rightarrow a=0\mbox{ and } b=0</math>

<math>{}_{}^{}\mbox{4. }V\in\mathbb{R}^n=
\left\lbrace\begin{pmatrix}\vdots\end{pmatrix}y,\qquad
e_1=\begin{pmatrix}1\\0\\\vdots\\0\end{pmatrix},
e_2=\begin{pmatrix}0\\1\\\vdots\\0\end{pmatrix},\ldots,
e_n=\begin{pmatrix}0\\0\\\vdots\\1\end{pmatrix}\right\rbrace</math>

:<math>{}_{}^{}e_1\ldots e_n\mbox{ are a basis of }V</math>
::<math>{}_{}^{}\mbox{They span }\begin{pmatrix}a_1\\\vdots\\a_n\end{pmatrix}=\sum a_ie_i</math>
::<math>{}_{}^{}\mbox{They are linearly independent. }\sum a_ie_i=0\Rightarrow \sum a_ie_i=
\begin{pmatrix}a_1\\\vdots\\a_n\end{pmatrix}=0\Rightarrow a_i=0 \quad\forall i</math>

<math>{}_{}^{}\mbox{5. In }V=P_3(\mathbb{R}),\qquad \beta=\lbrace 1,x,x^2,x^3\rbrace</math>

<math>{}_{}^{}\mbox{6. In }V=P_1(\mathbb{R})=\lbrace ax+b\rbrace,\qquad \beta=\lbrace 1+x,1-x\rbrace\mbox{ is a basis}</math>
:<math>{}_{}^{}\mbox{1. Generate }</math>
::<math>u_1+u_2=2\Rightarrow \frac{1}{2}(u_1+u_2)=1\mbox{ so }1 \in\mbox{ span }S</math>
::<math>u_1-u_2=2x\Rightarrow \frac{1}{2}(u_1-u_2)=x\mbox{ so }x \in\mbox{ span }S</math>
::<math>{}_{}^{}\mbox{ so span}\lbrace 1,x\rbrace \subset\mbox{ span }\beta</math>
:<math>{}_{}^{}\mbox{2. Linearly independent. Assume }au_1+bu_2=0</math>
::<math>\Rightarrow a(1+x)+b(1-x)=0\Rightarrow a+b+(a-b)x=0</math>
::<math>{}_{}^{}\Rightarrow a+b=0\mbox{ and }a-b=0</math>
::<math>(a+b)+(a-b)\Rightarrow 2a=0\Rightarrow a=0</math>
::<math>(a+b)-(a-b)\Rightarrow 2b=0\Rightarrow b=0</math>
<br>

<br>
<math>\mbox{Theorem}{}_{}^{}</math>

<math>{}_{}^{}\mbox{A subset }\beta\mbox{ of a vectorspace }V \mbox{ is a basis iff every }v\in V\mbox{ can be expressed as}</math>
<math>{}_{}^{}\mbox{a linear combination of elements in }</math>
<math>{}_{}^{}\beta \mbox{ in exactly one way.}</math>
<br>

<br>
<math>\mbox{Proof}{}_{}^{}</math>

<math>{}_{}^{}\mbox{It is a combination of things we already know.}</math>
#<math>{}_{}^{}\beta\mbox{ generates}</math>
#<math>{}_{}^{}\beta\mbox{ is linearly independent}</math>