Drorbn - User contributions [en]

09-240/Classnotes for Tuesday October 20

2009-10-21T00:41:20Z

Stephie:

== Definition ==
V & W are "isomorphic" if there exists a linear transformation T:V → W & S:W → V such that T∘S=IW and S∘T=IV

== Theorem ==
If V& W are field dimensions over F, then V is isomorphic to W iff dim V=dim W

== Corollary ==
If dim V = n then <math> \mathrm{V} \cong \mathrm{F^n} </math>
:Note: <math> \cong </math> represents isomorphism

Two "mathematical structures" are "isomorphic" if there's 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.

Ex:
The game of 15. Players alternate drawing one card each.
Goal: To have exactly three of your cards add to 15.

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

This game is isomorphic to Tic Tac Toe!

{| class="wikitable" border="1"
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| 7
|-
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| 1
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Converts to:

{| class="wikitable" border="1"
|-
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| 9
| X
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| O
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: S∘T=IV
: T∘S=IW
: T(OV)=OW

: T(x+y)=T(x)+T(y)
: T(cV)=cT(V)
: Likewise for <math> \mathrm{S} </math>

: z=x+y ⇒ T(z)=T(x)+T(y)
: u=7v ⇒ T(u)=7T(v)

Proof of Theorem <math> \Leftrightarrow </math> Assume dim V= dim W=n
: ∃ basis β= (U1...Un) of V
: α=(W1...Wn) of W
: by an earlier theorem, ∃ a l.t. T:V→W such that T(Ui)=Wi

(T(∑aiui)=∑aiT(ui)=∑aiui)

∃ a l.t. S:W→V s.t. S(Wi)=Ui

== Claim ==
: S∘T=Iv
: T∘S=Iw

== 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 ==
: 1. N(T=ker(T)={u∈V:Tu=0w}
: 2. R(T)=im(T)={T(u):u∈V}

== Prop/Def ==
: 1. N(T)'⊂V is a subspace of V-------nullity(T):=dim N(T)
: 2. 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)

09-240/Classnotes for Tuesday October 20

2009-10-20T23:13:29Z

Stephie:

09-240/Classnotes for Tuesday October 20

2009-10-20T22:25:54Z

Stephie:

== Def ==
V & W are "isomorphic" if there exists a linear transformation T:V → W & S:W → V such that T∘S=IW and S∘T=IV

== Theorem ==
If V& W are field dimensions over F, then V is isomorphic to W iff dim V=dim W

== Corollary ==
If dim V = n then <math> \mathrm{V} \cong \mathrm{F^n} </math>
:Note: <math> \cong </math> represents isomorphism

Two "mathematical structures" are "isomorphic" if there's 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.

Ex:
The game of 15. Players alternate drawing one card each.
Goal: To have exactly three of your cards add to 15.

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

This game is isomorphic to Tic Tac Toe!

{| class="wikitable" border="1"
|-
| 4
| 9
| 2
|-
| 3
| 5
| 7
|-
| 8
| 1
| 6
|}

Converts to:

{| class="wikitable" border="1"
|-
| O
| 9
| X
|-
| X
| O
| O
|-
| X
| X
| O
|}

Template:09-240/Navigation

2009-10-20T22:05:52Z

Stephie:

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|Sep 14
|[[09-240/Classnotes for Tuesday September 15|Tue]], [[09-240:HW1|HW1]], [[09-240:HW1 Solution|HW1 Solution]], [[09-240/Classnotes for Thursday September 17|Thu]]
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