12-240/Classnotes for Thursday October 18: Difference between revisions

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


== class note ==
== class note ==

Revision as of 11:58, 18 October 2012

DBN: Classes: 2012-13: 12-240 / Navigation
Additions to this web site no longer count towards good deed points.
# Week of... Notes and Links
1 Sep 10 About This Class, Tuesday, Thursday
2 Sep 17 HW1, Tuesday, Thursday, HW1 Solutions
3 Sep 24 HW2, Tuesday, Class Photo, Thursday
4 Oct 1 HW3, Tuesday, Thursday
5 Oct 8 HW4, Tuesday, Thursday
6 Oct 15 Tuesday, Thursday
7 Oct 22 HW5, Tuesday, Term Test was on Thursday. HW5 Solutions
8 Oct 29 Why LinAlg?, HW6, Tuesday, Thursday, Nov 4 is the last day to drop this class
9 Nov 5 Tuesday, Thursday
10 Nov 12 Monday-Tuesday is UofT November break, HW7, Thursday
11 Nov 19 HW8, Tuesday,Thursday
12 Nov 26 HW9, Tuesday , Thursday
13 Dec 3 Tuesday UofT Fall Semester ends Wednesday
F Dec 10 The Final Exam (time, place, style, office hours times)
Register of Good Deeds
12-240-ClassPhoto.jpg
Add your name / see who's in!
12-240-Splash.png

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?

12-240-DeckOfCards.png

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:

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 : N N, a A, a

class note

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    page1

  • page2

    page2

  • page3

    page3

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