12-240/Classnotes for Tuesday October 16

#	Week of...	Notes and Links
Additions to this web site no longer count towards good deed points.
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)
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Dror's notes above / Students' notes below

Theorems

1. If G generates, |G| $\geq \!\,$ n and G contains a basis, |G|=n then G is a basis

2. If L is linearly independent, |L| $\leq \!\,$ n and L can be extended to be a basis. |L|=n => L is a basis.

3.W $\subset \!\,$ V a subspace then W is finite dimensioned and dim W $\leq \!\,$ 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 : $\exists \!\,$ N $\in \!\,$ N, $\forall \!\,$ a $\in \!\,$ A, a $\leq \!\,$ N

has a maximal element: an element m $\in \!\,$ A, $\forall \!\,$ a $\in \!\,$ A, a $\leq \!\,$ m ( m + 1 $\notin \!\,$ A )