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Column Span and Rank

The column span (or column space) of a matrix $A$ is the span of its columns: the set of all vectors of the form $A\mathbf{x}$ as $\mathbf{x}$ ranges over $\mathbb{R}^n$ . It is a subspace of $\mathbb{R}^m$ .

The rank of $A$ is the dimension of its column span: $\text{Rank}(A) = \dim(\text{Col}(A))$ . Rank measures how many "truly independent directions" the matrix can produce as output.

Rank equals the number of pivot columns in the echelon form of $A$ . A matrix has full rank when its rank equals $\min(m, n)$ .

Formal View

Definition 3.5 — Column Space and Rank

The column space of

A \in \mathbb{R}^{m \times n}

\text{Col}(A) = \{A\mathbf{x} : \mathbf{x} \in \mathbb{R}^n\} \subseteq \mathbb{R}^m

. The rank is

\text{Rank}(A) = \dim(\text{Col}(A))

, equal to the number of pivots in echelon form.

Why This Matters

The column space is the set of all outputs a matrix can produce — it determines solvability.

Solvability of $A\mathbf{x} = \mathbf{b}$ : solvable iff $\mathbf{b} \in \text{Col}(A)$
In data science, rank determines how many "independent factors" explain a dataset
Rank of the Jacobian matrix determines degrees of freedom in robotic manipulators

Learning Resources

Column space, row space, null space

3Blue1Brown

Column space as the "reach" of a matrix.

10 min

Quiz

Question 1

Matrix $A$ has size $4 \times 3$ and rank 2. The column space has dimension:

Common Mistakes

Confusing rank with the number of rows or columns — rank is the dimension of the column space.
Thinking columns of $A$ directly form a basis for Col( $A$ ) — only the pivot columns do.