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09-240/Classnotes for Tuesday October 20

2009-12-14T13:56:13Z

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'''Definition''': '''V''' and '''W''' are "isomorphic" if there exist linear transformations <math>\mathrm{T : V \rightarrow W}</math> and <math>\mathrm{S : W \rightarrow V}</math> such that <math>\mathrm{T \circ S} = I_\mathrm{W}</math> and <math>\mathrm{S \circ T} = I_\mathrm{V}</math>

'''Theorem''': If '''V''' and '''W''' are finite-dimensional over ''F'', then '''V''' is isomorphic to '''W''' iff dim('''V''') = dim('''W''')

'''Corollary''': If dim('''V''') = ''n'' then <math>\mathrm{V} \cong F^n</math>
:Note: <math>\cong</math> represents "is isomorphic to"

----

Two "mathematical structures" are "isomorphic" if there exists a "bijection" between their elements which preserves all relevant relations between such elements.

'''Example''': Plastic chess is "isomorphic" to ivory chess, but it is not isomorphic to checkers.

'''Example''': The game of 15. Players alternate drawing one card each.

Goal: To have exactly three of your cards add to 15.

Sample game:
* X picks 3
* O picks 7
* X picks 8
* O picks ''4''
* X picks 1
* O picks ''6''
* X picks 2
* O picks ''5''
* 4 + 6 + 5 = 15. O wins.

This game is isomorphic to Tic Tac Toe!

{| class="wikitable" border="0" cellpadding="2" cellspacing="0"
|-
| style="border-style: none solid solid none" | 4
| style="border-style: none solid solid solid" | 9
| style="border-style: none none solid solid" | 2
|-
| style="border-style: solid solid solid none" | 3
| style="border-style: solid solid solid solid" | 5
| style="border-style: solid none solid solid" | 7
|-
| style="border-style: solid solid none none" | 8
| style="border-style: solid solid none solid" | 1
| style="border-style: solid none none solid" | 6
|}

: X: 3, 8, 1, 2
: O: 7, ''4'', ''6'', ''5'' -- Wins!

Converts to:

{| class="wikitable" border="0" cellpadding="2" cellspacing="0"
|-
| style="border-style: none solid solid none" | O
| style="border-style: none solid solid solid" | 9
| style="border-style: none none solid solid" | X
|-
| style="border-style: solid solid solid none" | X
| style="border-style: solid solid solid solid" | O
| style="border-style: solid none solid solid" | O
|-
| style="border-style: solid solid none none" | X
| style="border-style: solid solid none solid" | X
| style="border-style: solid none none solid" | O
|}

: <math>\mathrm{S \circ T} = I_\mathrm{V}</math>
: <math>\mathrm{T \circ S} = I_\mathrm{W}</math>
: <math>\mathrm T(O_\mathrm{V}) = O_\mathrm{W}</math>

: <math>\mathrm T(x + y) = T(x) + T(y)</math>
: <math>\mathrm T(cv) = c\mathrm T(v)</math>
: Likewise for <math>\mathrm S</math>

: <math>z = x + y \Rightarrow \mathrm T(z) = \mathrm T(x) + \mathrm T(y)</math>
: <math>u = 7v \Rightarrow \mathrm T(u) = 7\mathrm T(v)</math>

Proof of Theorem <math>\iff</math> Assume dim('''V''') = dim('''W''') = ''n''
: There exists basis <math>\beta = \{u_1, \ldots, u_n\} \in \mathrm V</math>
: <math>\alpha = \{w_1, ..., w_n\} \in \mathrm W</math>
: by an earlier theorem, there exists a l.t. <math>\mathrm{T : V \rightarrow W}</math> such that <math>\mathrm T(u_i) = w_i</math>

<math>\mathrm T(\sum a_i u_i) = \sum a_i \mathrm T(u_i) = \sum a_i w_i</math>

There exists a l.t. <math>\mathrm{S : W \rightarrow V}</math> such that <math>\mathrm S(w_i) = u_i</math>

== Claim ==
: <math>\mathrm{S \circ T} = I_\mathrm{V}</math>
: <math>\mathrm{T \circ S} = I_\mathrm{W}</math>

== Proof ==
If u∈<math> \mathrm{V} </math> unto U=∑aiui
: (S∘T)(u)=S(T(u))=S(T(∑aiui))
: =S(∑aiwi)=∑aiui=u
: ⇒S∘T=Iv...
: ⇒Assume T&S as above exist
: Choose a basis β= (U1...Un) of V

== Claim ==
α=(W1=Tu1, W2=Tu2, ..., Wn=Tun)
: is a basis of W, so dim W=n

== Proof ==
α is lin. indep.
: T(0)=0=∑aiwi=∑aiTui=T(∑aiui)
: Apply S to both sides:
: 0=∑aiui
: So ∃iai=0 as β is a basis

α Spans W
: Given any w∈W let u=S(W)
: As β is a basis find ais in F s.t. v=∑aiui
Apply T to both sides: T(S(W))=T(u)=T(∑aiui)=∑aiT(ui)=∑aiWi
::: ∴ I win!!! (QED)

: T T
: V → W ⇔ V' → W'
: rank T=rank T'
Fix t:V→Wa l.t.<math>Insert formula here</math>

== Definition ==
# N(T) = ker(T) = {u∈V : Tu = 0W}
# R(T) = im(T) = {T(u) : u∈V}

== Prop/Def ==
# N(T) ⊂ V is a subspace of V-------nullity(T) := dim N(T)
# R(T) ⊂ W is a subspace of W--------rank(T) := dim R(T)

== Proof 1 ==
: x,y ∈N(T)⇒T(x)=0, T(y)=0
: T(x+y)=T9x)+T(y)=0+0=0
: x+y∈N(T)
::: ∴ I win!!! (QED)

== Proof 2 ==
: Let y∈R(T)⇒fix x s.t y=T(x),
: --------7y=7T(x)=T(7x)
: ----------⇒7y∈R(T)
::: ∴ I win!!! (QED)

== Examples ==
1.
: 0:V→W---------N(0)=V
: R(0)={0W}-----------nullity(0)=dim V
: --------------rank(0)=0
:: dim V+0=dimV
2.
:IV:V→V
:N(I)={0}
:nullity=0
:R(I)=dim V
:2'If T:V→W is an imorphism
:N(T)={0}
:nullity =0
:R(T)=W
:rank=dim W
::0+dim V=dim V
3.
:D:P7(R)→P7(R)
:Df=f'
::N(D)={C⊃C°: C∈R}=P0(R)
:R(D)⊂P6(R)
::nullity(D)=1
::basis:(1x°)
::rank(D)=7
:::7+1=8
4.
:3':D2:P7(R)
:D2f=f''
:W(D2)={ax+b: a,b∈R}=P1(R)
::nullity(D2)=2
::R(D2)=P5(R)
:::rank (D2)=6
::6+2=8

== Theorem ==
(rank-nullity Theorem, a.k.a. dimension Theorem)
:nullity(T)+rank(T)=dim V
:(for a l.t. T:V→W) when V is F.d.

== Proof ==
(To be continued next day)

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09-240/Classnotes for Tuesday October 20

2009-12-14T13:49:47Z

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09-240/Classnotes for Thursday October 15

2009-12-14T13:47:47Z

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===A Further Set of Lecture Notes===

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===Nevena's Tutorial (October 15th)===

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09-240/Classnotes for Thursday October 15

2009-12-14T13:44:25Z

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Nevena's Tutorial (October 15th)

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09-240/Classnotes for Thursday October 15

2009-12-14T13:37:07Z

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Nevena's Tutorial (October 15th)

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09-240/Classnotes for Thursday October 1

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==A Message from Accessibility Services==
:"Accessibility Services requires dependable volunteer note-takers in this course to assist students with disabilities. Those who are interested in assisting with this essential service will gain valuable volunteer experience and a certificate of recognition. If you are interested in becoming a volunteer note-taker, please take an information form and register online, or visit the Accessibility Services office at 215 Huron Street."

You can also sign up online, at http://www.accessibility.utoronto.ca/newreturn/note_taking_accommodation.htm.

{{09-240/Class Notes Warning}}
==Class notes for today==
<gallery>
Image:09-240_OCT01_NOTES.JPG
Image:ALA240-2009 - October 1st.pdf|A complete copy of notes for the lecture given on October 1st by Professor Natan (in PDF format)
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(In the above gallery, there is a complete copy of notes for the lecture given on November 10th by Professor Natan (in PDF format).

 

Nevena's Tutorial (October 1st)

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09-240/Classnotes for Thursday October 8

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A complete set of notes for the lecture given by Professor Natan on October 8th is included in the above gallery (in PDF form).

Nevena's Tutorial (October 8th):

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09-240

2009-12-12T23:07:57Z

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==Algebra I==
===Department of Mathematics, University of Toronto, Fall 2009===

{{09-240/Crucial Information}}

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

===Further Resources===

* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]
* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions].
* [http://www.math.toronto.edu/murnaghan/courses/mat240/index.html Last year's MAT240 web site].
* [[06-240|My 2006 Math 240 web site]].
* [http://www.math.toronto.edu/~megumi/MAT240/240.html The 2005 MAT240 site].

===Interesting Links===

* [http://aix1.uottawa.ca/~jkhoury/app.htm Interesting Linear Algebra Applications]
* [http://linear.ups.edu/jsmath/latest/fcla-jsmath-latest.html Free Alternative Online Linear Algebra Textbook]
* [http://www.rose-hulman.edu/~bryan/googleFinalVersionFixed.pdf "The $25,000,000,000 Eigenvector" - The first half and parts of the second half are accessible given only MAT240.]

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2009-12-10T19:00:02Z

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|[[09-240/Final Exam Preparation Forum|Forum]], [[09-240/On The Final Exam|Office Hours]]
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|[[09-240/The Final Exam|Final on Dec 16]]
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09-240/Classnotes for Tuesday December 1

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--- Wiki Format ---

MAT240 – December 1st

Basic Properties of det: Mnxn→F: 0 det(I) = 1

1. <math>det(E'_{i,j\,\!}A) = -det(A) ; |E'_{i,j\,\!}|= -1. [Note: det(EA) = |E||A|]</math>

* Also, note that exchanging two rows flips the sign.

2. <math>det(E^2_{i,c\,\!}A) = det(A) ; |E^2_{i,j,c\,\!}| = 1</math>

* These are "enough"!

3. <math>det((E_{i,j,c\,\!}A) = det(A) ; |E^3_{i,j,c\,\!}| = 1</math>

* Adding a multiple of one row to another does not change the determinant.

The determinant of any matrix can be calculated using the properties above.

Theorem:

If <math> det' : M_{nxn\,\!}</math>→F satisfies properties 0-3 above, then <math>det' = det</math>

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

Philosophical remark: Why not begin our inquiry with the properties above?

We must find an implied need for their use; thus, we must know whether a function <math>det</math> exists first.