Theorem 9

If a vector space V has a basis $\beta =${$\vec{b}_1,...,\vec{b}_n$} then any set of vectors from V containing more than n vectors must be linearly dependent

Theorem 10

If a vectorspace has a basis of n - vectors then every $\beta$ of the vectorspace has n - vectors

note |…| means number of elements
Let $|\beta_1| = n$
and $|\beta_2| >n$. By Theorem 9, we can see that $\beta_2$ will not be linearly independent therefore cannot be a basis by the definition of a basis.

if $|\beta_2| < n$, then this "basis" wouldn't have a pivot in every column and therefore would not span $\Re^n$

Therefore, for a new basis to truly be a basis for the same dimension as another, it must have the same number of elements in its column space (or n number of vectors).

Theorem 11

Let H be a subspace of a finite dimensional vectors space. Any linearly independent set in H can be expanded to a basis of H

Theorem 12 - Basis Theorem

Let V be a p-dimensional vectors space where $p \geq 1$

  1. Any linearly independent set of p vectors is a basis
  2. Any p-vectors that span V is a basis
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