__Dimension of a Subspace__
The dimension of a non-zero subspace

$S$ (usually denoted as dim

$S$), is the # of vectors in any basis for

$S$. Since the null space and column space is a subspace, we can find their dimensions.

Note: Dimension of the column space = rank

__Finding the Rank of a matrix:__
1. Find the basis for the column space

2. Count the # of vectors in the basis. That is the rank.

Shortcut: Count the # of pivots in the matrix

__Finding the dimensions of the null space:__
1. Find the basis for the null space

2. Count the # of vectors in the basis. That is the dimension.

Shortcut: Count the # of free variables in the matrix.

__The Rank Theorem__
If a matrix

$A$ has

$n$ columns, then rank

$A+$ dim

$N(A) = n$.