Dimension and rank  Subspace of \(\Bbb{R}^n\)
Dimension and rank
Lessons
Notes:
Dimension of a Subspace
The dimension of a nonzero 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 $N(A) = n$.

Intro Lesson
Dimension and Rank Overview: