12240/Classnotes for Tuesday October 16
From Drorbn

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 N
has a maximal element: an element m A, a A, a m ( m + 1 A )