09-240/Classnotes for Tuesday October 20 - Revision history

Mantynen at 13:56, 14 December 2009

2009-12-14T13:56:13Z

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Mantynen at 13:49, 14 December 2009

2009-12-14T13:49:47Z

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Bright at 00:02, 11 December 2009

2009-12-11T00:02:59Z

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

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C8sd: Put second player on bottom.

2009-12-07T07:46:53Z

Put second player on bottom.

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

	Converts to:		Converts to:

C8sd: Add standard templates.

2009-12-07T07:25:41Z

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			{{09-240/Navigation}}
			{{09-240/Class Notes Warning}}


	'''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>		'''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>

C8sd: Missed closing tag.

2009-12-07T07:24:31Z

Missed closing tag.

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

	Proof of Theorem <math>\iff</math> Assume dim('''V''') = dim('''W''') = ''n''		Proof of Theorem <math>\iff</math> Assume dim('''V''') = dim('''W''') = ''n''

C8sd: Format first half.

2009-12-07T07:20:41Z

Format first half.

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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>
	== Definition ==
	V & W are "isomorphic" if there exists a linear transformation T:V → W & S:W → V such that T∘S=I<sub>W</sub> and S∘T=I<sub>V</sub>

			'''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>
	== Theorem ==
		⚫	:Note: <math>\cong</math> represents "is isomorphic to"
	If V& W are field dimensions over F, then V is isomorphic to W iff dim V=dim W

			----

		⚫	Two "mathematical structures" are "isomorphic" if there exists a "bijection" between their elements which preserves all relevant relations between such elements.
	== Corollary ==
⚫	If dim V = n then <math> \mathrm{V} \cong ~~\mathrm{~~F^n} </math>
⚫	:Note: <math> \cong </math> represents ~~isomorphism~~

		⚫	'''Example''': Plastic chess is "isomorphic" to ivory chess, but it is not isomorphic to checkers.
⚫	Two "mathematical structures" are "isomorphic" if there's a "bijection" between their elements which preserves all relevant relations between such elements.

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

			Sample game:
⚫	O: 7, ''4, 6, 5'' → Wins!
			* X picks 3
⚫	X: 3, 8, 1, 2
			* 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!		This game is isomorphic to Tic Tac Toe!
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	\| style="border-style: solid none none solid" \| 6		\| style="border-style: solid none none solid" \| 6
	\|}		\|}

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

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

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

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

	Proof of Theorem <math> \~~Leftrightarrow~~ </math> Assume dim V= dim W=n		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>
	: ∃ basis β= (U<sub>1</sub>...U<sub>n</sub>) of V
			: <math>\alpha = \{w_1, ..., w_n\} \in \mathrm W</math>
	: α=(W<sub>1</sub>...W<sub>n</sub>) of W
	: by an earlier theorem, ∃ a l.t. T:~~V→W~~ such that ~~T(U~~<~~sub>i</sub~~>)=~~W<sub>i~~</~~sub~~>		: 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>
	(T(∑a<sub>i</sub>u<sub>i</sub>)=∑a<sub>i</sub>T(u<sub>i</sub>)=∑a<sub>i</sub>u<sub>i</sub>)

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


	== Claim ==		== Claim ==
			: <math>\mathrm{S \circ T} = I_\mathrm{V}</math>
	: S∘T=I<sub>v</sub>
			: <math>\mathrm{T \circ S} = I_\mathrm{W}</math>
	: T∘S=I<sub>w</sub>

C8sd: Format.

2009-12-07T06:50:32Z

Format.

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	This game is isomorphic to Tic Tac Toe!		This game is isomorphic to Tic Tac Toe!

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

	Converts to:		Converts to:

	{\| class="wikitable" border="1"		{\| class="wikitable" border="0" cellpadding="2" cellspacing="0"
	\|-		\|-
			\| style="border-style: none solid solid none" \| O
	\| O
			\| style="border-style: none solid solid solid" \| 9
	\| 9
			\| style="border-style: none none solid solid" \| X
	\| X
	\|-		\|-
			\| style="border-style: solid solid solid none" \| X
	\| X
			\| style="border-style: solid solid solid solid" \| O
	\| O
			\| style="border-style: solid none solid solid" \| O
	\| O
	\|-		\|-
			\| style="border-style: solid solid none none" \| X
	\| X
			\| style="border-style: solid solid none solid" \| X
	\| X
			\| style="border-style: solid none none solid" \| O
	\| O
	\|}		\|}

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	== Definition ==		== Definition ==
	~~: 1.~~ N(T=ker(T)={u∈V:Tu=0<sub>w</sub>}		# N(T) = ker(T) = {u∈V : Tu = 0<sub>W</sub>}
	~~: 2.~~ R(T)=<sub>i</sub>m(T)={T(u):u∈V}		# R(T) = <sub>i</sub>m(T) = {T(u) : u∈V}

	== Prop/Def ==		== Prop/Def ==
	~~: 1.~~ N(T)~~'⊂V~~ is a subspace of V-------nullity(T):=dim N(T)		# 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)		# R(T) ⊂ W is a subspace of W--------rank(T) := dim R(T)

Stephie at 00:41, 21 October 2009

2009-10-21T00:41:20Z

@@ Line 105: / Line 105: @@
 : Given any w∈W let u=S(W)
 : As β is a basis find a<sub>i</sub>s in F s.t. v=∑a<sub>i</sub>u<sub>i</sub>
-Apply T to both sides: T(S(W))=T(u)=T(∑a<sub>i</sub>u<sub>i</sub>)=∑a<sub>i</sub>T(u<sub>i</sub>)=∑a<sub>i</sub>W<sub>i</sub>  ∴ I win!!! (QED)
+Apply T to both sides: T(S(W))=T(u)=T(∑a<sub>i</sub>u<sub>i</sub>)=∑a<sub>i</sub>T(u<sub>i</sub>)=∑a<sub>i</sub>W<sub>i</sub>

Stephie at 23:13, 20 October 2009

2009-10-20T23:13:29Z

@@ Line 1: / Line 1: @@
-== Def ==
+== Definition ==
 V & W are "isomorphic" if there exists a linear transformation T:V → W & S:W → V such that T∘S=I<sub>W</sub>  and S∘T=I<sub>V</sub>
@@ Line 56: / Line 56: @@
 |   O
 |}

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