Difference between revisions of "12-240/Classnotes for Thursday October 18"
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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. | 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 | + | 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]] | [[Image:12-240-DeckOfCards.png|center]] | ||
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{{12-240:Dror/Students Divider}} | {{12-240:Dror/Students Divider}} | ||
− | |||
+ | ===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 <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:''' | ||
+ | |||
+ | 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 [[User:starash|starash]]== | ||
+ | |||
+ | <gallery> | ||
+ | Image:12-240-1018-1.jpg |page1 | ||
+ | Image:12-240-1018-2.jpg |page2 | ||
+ | </gallery> | ||
+ | |||
+ | == Lecture notes uploaded by [[User:Grace.zhu|gracez]] == | ||
<gallery> | <gallery> | ||
− | Image:12-240- | + | Image:12-240-O18-1.jpg|Page 1 |
− | Image:12-240- | + | Image:12-240-O18-2.jpg|Page 2 |
</gallery> | </gallery> |
Latest revision as of 21:40, 12 December 2012
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Contents |
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)