# 12-240/Classnotes for Tuesday October 16

 Dror's notes above / Students' notes below

## Theorems

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

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

3.W ${\displaystyle \subset \!\,}$ V a subspace then W is finite dimensioned and dim W ${\displaystyle \leq \!\,}$ 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.

2. Bounded : ${\displaystyle \exists \!\,}$ N ${\displaystyle \in \!\,}$ N, ${\displaystyle \forall \!\,}$ a ${\displaystyle \in \!\,}$ A, a ${\displaystyle \leq \!\,}$ N
has a maximal element: an element m ${\displaystyle \in \!\,}$ A, ${\displaystyle \forall \!\,}$ a ${\displaystyle \in \!\,}$ A, a ${\displaystyle \leq \!\,}$ m ( m + 1 ${\displaystyle \notin \!\,}$ A )