# 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.

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 )