Column Space of a Matrix

Definition

The column space of an $m \times n$ matrix $A$, written $\operatorname{Col}(A)$, is the span of its column vectors: $\operatorname{Col}(A) = \{A \mathbf{x} : \mathbf{x} \in \mathbb{R}^n\} \subseteq \mathbb{R}^m$

It's a subspace of $\mathbb{R}^m$ (the output space). Its dimension is $\operatorname{rank}(A)$.

Key interpretation: $\operatorname{Col}(A)$ is exactly the set of vectors $\mathbf{b}$ for which $A \mathbf{x} = \mathbf{b}$ is consistent (has a solution). If $\mathbf{b} \notin \operatorname{Col}(A)$, no $\mathbf{x}$ can produce it.

Step-by-step

$A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 1 & 1 \end{pmatrix}$ (3 × 2 matrix; output space is $\mathbb{R}^3$).

Step 1 — Are the columns independent?

Columns: $\mathbf{a}_1 = (1, 0, 1)^T$, $\mathbf{a}_2 = (0, 1, 1)^T$.

Set $c_1 \mathbf{a}_1 + c_2 \mathbf{a}_2 = \mathbf{0}$:

• $c_1 = 0$ (first component),
• $c_2 = 0$ (second component),
• $c_1 + c_2 = 0$ (third component).

Only the trivial solution — independent.

Step 2 — A basis for $\operatorname{Col}(A)$.

Since both columns are independent, the basis is $\{(1, 0, 1)^T, (0, 1, 1)^T\}$.

$\operatorname{rank}(A) = 2, \quad \operatorname{Col}(A) \subseteq \mathbb{R}^3 \text{ is 2-dimensional.}$

Step 3 — Describe the column space as an equation.

Any $\mathbf{b} \in \operatorname{Col}(A)$ satisfies $\mathbf{b} = c_1 \mathbf{a}_1 + c_2 \mathbf{a}_2$, i.e. $\mathbf{b} = \begin{pmatrix} c_1 \\ c_2 \\ c_1 + c_2 \end{pmatrix}$

So $b_3 = b_1 + b_2$. The column space is the plane $\{(b_1, b_2, b_3) : b_3 = b_1 + b_2\}$ in $\mathbb{R}^3$.

Geometric picture

In $\mathbb{R}^3$, we have a 2D plane through the origin with normal vector $(1, 1, -1)$ (from $b_1 + b_2 - b_3 = 0$). Only target vectors $\mathbf{b}$ on this plane can be produced by $A \mathbf{x}$.

• $\mathbf{b} = (1, 2, 3)$? Check: $1 + 2 = 3$ ✓. Solvable.
• $\mathbf{b} = (1, 2, 5)$? Check: $1 + 2 = 3 \ne 5$ ✗. Not solvable.

Why the column space matters

• Consistency test: $A \mathbf{x} = \mathbf{b}$ has a solution ⟺ $\mathbf{b} \in \operatorname{Col}(A)$.
• Surjectivity: $A$ is onto $\mathbb{R}^m$ ⟺ $\operatorname{Col}(A) = \mathbb{R}^m$ ⟺ $\operatorname{rank}(A) = m$.
• Least squares: when $\mathbf{b} \notin \operatorname{Col}(A)$, we project $\mathbf{b}$ onto $\operatorname{Col}(A)$ to find the closest fittable target.

Column space vs. row space

Both have the same dimension: $\operatorname{rank}(A) = \dim \operatorname{Col}(A) = \dim \operatorname{Row}(A)$.

They live in different ambient spaces though: $\operatorname{Col}(A) \subseteq \mathbb{R}^m$, $\operatorname{Row}(A) \subseteq \mathbb{R}^n$. And they're related by $\operatorname{Col}(A) = \operatorname{Row}(A^T)$.

Common mistakes

• Using RREF columns as the basis. Row reduction changes the column space! Use original columns in pivot positions (identified by RREF) for the basis.
• Confusing column space with column rank. The space is a set of vectors; the rank is its dimension.
• Testing $\mathbf{b}$ wrong. To check if $\mathbf{b} \in \operatorname{Col}(A)$, augment $[A \mid \mathbf{b}]$ and row reduce; if the bottom row becomes $[0 \cdots 0 \mid c]$ with $c \ne 0$, then $\mathbf{b} \notin \operatorname{Col}(A)$.

Try it in the visualization

A 3D plot shows the two column vectors as arrows from the origin. They span a plane — the column space. Drag a test point $\mathbf{b}$ around; it turns green when it lies on the plane (solvable) and red otherwise.