12-240/Classnotes for Thursday October 18: Difference between revisions
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== Theorems == |
== Theorems == |
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1. If G generates, |G| <math>\ge \!\,</math> n and G contains a |
1. If G generates, |G| <math>\ge \!\,</math> n and G contains a basis, |G|=n then G is a basis |
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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. |
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3.W <math>\subset \!\,</math> V a subspace then W is finite dimensioned and dim W <math>\le \!\,</math> dim V |
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If dim W = dim V, then V = W |
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If dim W < dim V, then any basis of W can be extended to be a basis of V |
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Proof of W is finite dimensioned: |
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== class note == |
== class note == |
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Revision as of 11:54, 18 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 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?
See also a video and the transcript of that video.
| Dror's notes above / Students' notes below |
Theorems
1. If G generates, |G| n and G contains a basis, |G|=n then G is a basis
2. If L is linearly independent, |L| n and L can be extended to be a basis. |L|=n => L is a basis.
3.W V a subspace then W is finite dimensioned and dim W 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:
class note
page1
page2
page3
