12-240/Classnotes for Thursday October 18 - Revision history

Drorbn at 01:40, 13 December 2012

2012-12-13T01:40:52Z

← Older revision		Revision as of 21:40, 12 December 2012
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			===Riddle Along===
⚫	~~'''~~== Linear transformation ==~~'''~~
			The game of 15 is played as follows. Two players alternate choosing cards numbered between 1 and 9, with repetitions forbidden, so that the game ends at most after 9 moves (or four and a half rounds). The first player to have within her/his cards a set of precisely 3 cards that add up to 15 wins.

			Does this game have a winning strategy? What is it? Who wins, the first to move or the second? Why am I asking this question at this particular time?

			[[Image:12-240-DeckOfCards.png\|center]]

			See also [https://media.library.utoronto.ca/play.php?DJ6CPFxByy2J&id=8503 a video] and the [https://cmc.math.ca/home/videos/game-of-15-and-isomorphisms/ transcript] of that video.

			{{12-240:Dror/Students Divider}}

		⚫	===Linear transformation===

	'''Definition:'''		'''Definition:'''

Yaaleni.vijay at 19:29, 8 November 2012

2012-11-08T19:29:01Z

← Older revision		Revision as of 15:29, 8 November 2012
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	'''== Linear transformation =='''		'''== Linear transformation =='''

⚫	~~'''Definition:'''~~ A function L: V-> W is called a linear transformation if it preserve following structures:
			'''Definition:'''

		⚫	A function L: V-> W is called a linear transformation if it preserve following structures:

	1) L(x + y)= L(x) + L(y)		1) L(x + y)= L(x) + L(y)
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	2) L: V -> W is a linear transformation iff <math>\forall\,\!</math> c <math>\in\,\!</math> F, <math>\forall\,\!</math> x, y <math>\in\,\!</math> V: L(cx + y)= cL(x) + L(y)		2) L: V -> W is a linear transformation iff <math>\forall\,\!</math> c <math>\in\,\!</math> F, <math>\forall\,\!</math> x, y <math>\in\,\!</math> V: L(cx + y)= cL(x) + L(y)

	'''Proof:		'''Proof:'''

	'''
	1) take c= 0 in F and x=0 in V. Then L(cx)=cL(x) -> L(0 of F * 0 of V)=(0 of F)*L(0 of V)=0 of W		1) take c= 0 in F and x=0 in V. Then L(cx)=cL(x) -> L(0 of F * 0 of V)=(0 of F)*L(0 of V)=0 of W

	2)(=>)Assume L is linear transformation		2)(=>)Assume L is linear transformation

	L(cx + y)= L(cx) + L(y)= c*L(x) + L(y)		L(cx + y)= L(cx) + L(y)= c*L(x) + L(y)

			(<=) 1. Follows from L(cx+y) = cL(x)+L(y) by taking c=1

	~~(<=) 1~~. Follows ~~from L(cx+y) = cL(x)+L(y)~~ by taking c=1		2. Follows by taking y=0
	2. Follows by taking y=0

	'''Examples'''		'''Examples'''

Grace.zhu: /* lecture note on oct 18, uploaded by starash */

2012-11-04T18:16:27Z

lecture note on oct 18, uploaded by starash

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	Image:12-240-1018-2.jpg \|page2		Image:12-240-1018-2.jpg \|page2
			</gallery>

			== Lecture notes uploaded by [[User:Grace.zhu\|gracez]] ==
			<gallery>
			Image:12-240-O18-1.jpg\|Page 1
			Image:12-240-O18-2.jpg\|Page 2
	</gallery>		</gallery>

Yaaleni.vijay: /* Linear transformation */

2012-11-02T18:31:48Z

Linear transformation

@@ Line 1: / Line 1: @@
 {{12-240/Navigation}}
-== Linear transformation ==
+'''== Linear transformation =='''
-Definition: A function L: V-> W is called a linear transformation if it preserve following structures:
+'''Definition:''' A function L: V-> W is called a linear transformation if it preserve following structures:
 ) L(x + y)= L(x) + L(y)
 ) L(cx)= c.L(x)
 ) L(0 of V) = 0 of W
-Proposition:
+'''Proposition:'''
 == lecture note on oct 18, uploaded by [[User:starash|starash]]==

Dangkhoa.nguyen: /* Linear transformation */

2012-10-30T03:49:12Z

Linear transformation

← Older revision		Revision as of 23:49, 29 October 2012
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	1) property 2 leads to property 3		1) property 2 leads to property 3
	2) L: V -> W is a linear transformation iff <math>\forall\,\!</math> c <math>\in\,\!</math>, <math>\forall\,\!</math> x, y <math>\in\,\!</math> V: L(cx + y)= cL(x) + L(y)		2) L: V -> W is a linear transformation iff <math>\forall\,\!</math> c <math>\in\,\!</math>, <math>\forall\,\!</math> x, y <math>\in\,\!</math> V: L(cx + y)= cL(x) + L(y)
			Proof:
			1) take c= 0 in F and x=0 in V. Then L(cx)=cL(x) -> L(0 of F . 0 of V)=(0 of F).L(0 of V)=0 of W
			2) L(cx + y)= L(cx) + L(y)= c.L(x) + L(y)

	== lecture note on oct 18, uploaded by [[User:starash\|starash]]==		== lecture note on oct 18, uploaded by [[User:starash\|starash]]==

Dangkhoa.nguyen: /* Linear transformation */

2012-10-30T03:38:19Z

Linear transformation

← Older revision		Revision as of 23:38, 29 October 2012
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	2) L(cx)= c.L(x)		2) L(cx)= c.L(x)
	3) L(0 of V) = 0 of W		3) L(0 of V) = 0 of W

			Proposition:

			1) property 2 leads to property 3
			2) L: V -> W is a linear transformation iff <math>\forall\,\!</math> c <math>\in\,\!</math>, <math>\forall\,\!</math> x, y <math>\in\,\!</math> V: L(cx + y)= cL(x) + L(y)

	== lecture note on oct 18, uploaded by [[User:starash\|starash]]==		== lecture note on oct 18, uploaded by [[User:starash\|starash]]==

Dangkhoa.nguyen: /* Linear transformation */

2012-10-26T02:49:16Z

Linear transformation

← Older revision		Revision as of 22:49, 25 October 2012
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	== Linear transformation ==		== Linear transformation ==
			Definition: A function L: V-> W is called a linear transformation if it preserve following structures:

			1) L(x + y)= L(x) + L(y)
			2) L(cx)= c.L(x)
			3) L(0 of V) = 0 of W

	== lecture note on oct 18, uploaded by [[User:starash\|starash]]==		== lecture note on oct 18, uploaded by [[User:starash\|starash]]==

Dangkhoa.nguyen at 02:43, 26 October 2012

2012-10-26T02:43:54Z

← Older revision		Revision as of 22:43, 25 October 2012
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			== Linear transformation ==

	== lecture note on oct 18, uploaded by [[User:starash\|starash]]==		== lecture note on oct 18, uploaded by [[User:starash\|starash]]==

Drorbn at 10:57, 25 October 2012

2012-10-25T10:57:42Z

← Older revision		Revision as of 06:57, 25 October 2012
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	== lecture note on oct 18, uploaded by [[User:starash\|starash]]==		== lecture note on oct 18, uploaded by [[User:starash\|starash]]==

Zetalda at 00:20, 25 October 2012

2012-10-25T00:20:00Z

← Older revision		Revision as of 20:20, 24 October 2012
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	===Riddle Along===
	The game of 15 is played as follows. Two players alternate choosing cards numbered between 1 and 9, with repetitions forbidden, so that the game ends at most after 9 moves (or four and a half rounds). The first player to have within her/his cards a set of precisely 3 cards that add up to 15 wins.

	Does this game have a winning strategy? What is it? Who wins, the first to move or the second? Why am I asking this question at this particular time?

	[[Image:12-240-DeckOfCards.png\|center]]

	See also [https://media.library.utoronto.ca/play.php?DJ6CPFxByy2J&id=8503 a video] and the [https://cmc.math.ca/home/videos/game-of-15-and-isomorphisms/ transcript] of that video.

	{{12-240:Dror/Students Divider}}

	== Theorems ==
	1. If G generates, \|G\| <math>\ge \!\,</math> n and G contains a basis, \|G\|=n then G is a basis

	2. If L is linearly independent, \|L\| <math>\le \!\,</math> n and L can be extended to be a basis. \|L\|=n => L is a basis.

	3.W <math>\subset \!\,</math> V a subspace then W is finite dimensioned and dim W <math>\le \!\,</math> dim V

	If dim W = dim V, then V = W
	If dim W < dim V, then any basis of W can be extended to be a basis of V

	Proof of W is finite dimensioned:

	Let L be a linearly independent subset of W which is of maximal size.

	Fact about '''N'''
	: Every subset A of '''N''', which is:

	1. Non empty

	2. Bounded : <math>\exist \!\,</math> N <math>\in \!\,</math> '''N''', <math>\forall \!\,</math> a <math>\in \!\,</math> A, a <math>\le \!\,</math> N

	has a maximal element: an element m <math>\in \!\,</math> A, <math>\forall\!\,</math> a <math>\in \!\,</math> A, a <math>\le \!\,</math> m ( m + 1 <math>\notin \!\,</math> A )

	== class note ==


	<gallery>
	Image:12-240-Oct-15-Page-1.jpg \|page1
	Image:12-240-Oct-15-Page-2.jpg \|page2
	Image:12-240-Oct-15-Page-3.jpg \|page3
	</gallery>


	== lecture note on oct 18, uploaded by [[User:starash\|starash]]==		== lecture note on oct 18, uploaded by [[User:starash\|starash]]==