Dimension of a Subspace
The dimension of a non-zero subspace
(usually denoted as dim
), is the # of vectors in any basis for
. 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
columns, then rank