12-240/Classnotes for Thursday October 18: Difference between revisions
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===Riddle Along=== |
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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. |
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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? |
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[[Image:12-240-DeckOfCards.png|center]] |
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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. |
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{{12-240:Dror/Students Divider}} |
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'''Definition:''' |
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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) |
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'''Proof: |
'''Proof:''' |
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''' |
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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 |
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2)(=>)Assume L is linear transformation |
2)(=>)Assume L is linear transformation |
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L(cx + y)= L(cx) + L(y)= c*L(x) + L(y) |
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(<=) 1. Follows from L(c*x+y) = c*L(x)+L(y) by taking c=1 |
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2. Follows by taking y=0 |
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2. Follows by taking y=0 |
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'''Examples''' |
'''Examples''' |
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Latest revision as of 20:40, 12 December 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?
See also a video and the transcript of that video.
| Dror's notes above / Students' notes below |
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
Proposition:
1) property 2 => property 3
2) L: V -> W is a linear transformation iff c F, x, y 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)(=>)Assume L is linear transformation
L(cx + y)= L(cx) + L(y)= c*L(x) + L(y)
(<=) 1. Follows from L(c*x+y) = c*L(x)+L(y) by taking c=1
2. Follows by taking y=0
Examples
1. L: R^2 -> R^2 by
2. P,Q: P(F)
lecture note on oct 18, uploaded by starash
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Lecture notes uploaded by gracez
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