12-240/Classnotes for Tuesday October 16

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Dror's notes above / Students' notes below

Theorems

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

2. If L is linearly independent, |L| \le \!\, 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 \le \!\, 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 : \exist \!\, N \in \!\, N, \forall \!\, a \in \!\, A, a \le \!\, N

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

class note