Column Space

The column space $C(A)$ is the set of all linear combinations of the columns of $A$.

$$C(A) = \{A\mathbf{x} : \mathbf{x} \in \mathbb{R}^n\}$$

Key Facts

• $C(A)$ is a subspace of $\mathbb{R}^m$ (for $m \times n$ matrix $A$)
• $Ax = b$ has a solution if and only if $b \in C(A)$
• The dimension of $C(A) = \text{rank}(A)$ = number of pivot columns

How to Find It

1. Do elimination on $A$
2. The pivot columns of the original $A$ (not the echelon form!) form a basis for $C(A)$