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 has 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?
+
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}}
== class note ==
 
  
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===Linear transformation===
  
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'''Definition:'''
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A function L: V-> W is called a linear transformation if it preserve following structures:
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1) L(x + y)= L(x) + L(y)
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2) L(cx)= c.L(x)
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3) L(0 of V) = 0 of W
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'''Proposition:'''
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1) property 2 => property 3
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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)
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'''Proof:'''
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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
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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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'''Examples'''
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1. L: '''R'''^2 -> '''R'''^2 by
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2. P,Q: P(F)
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== lecture note on oct 18, uploaded by [[User:starash|starash]]==
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<gallery>
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Image:12-240-1018-1.jpg |page1
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Image:12-240-1018-2.jpg |page2
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</gallery>
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== Lecture notes uploaded by [[User:Grace.zhu|gracez]] ==
 
<gallery>
 
<gallery>
Image:12-240-Oct-15-Page-1.jpg |page1
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Image:12-240-O18-1.jpg|Page 1
Image:12-240-Oct-15-Page-2.jpg |page2
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Image:12-240-O18-2.jpg|Page 2
 
</gallery>
 
</gallery>

Latest revision as of 21:40, 12 December 2012

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?

12-240-DeckOfCards.png

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 \forall\,\! c \in\,\! F, \forall\,\! x, y \in\,\! 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

Lecture notes uploaded by gracez