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Column Space — Definition, Formula & Examples

The column space of a matrix is the set of all possible linear combinations of its column vectors. It tells you exactly which output vectors b\mathbf{b} can be reached by the equation Ax=bA\mathbf{x} = \mathbf{b}.

Given an m×nm \times n matrix AA with columns a1,a2,…,an∈Rm\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n \in \mathbb{R}^m, the column space Col(A)\text{Col}(A) is the subspace of Rm\mathbb{R}^m spanned by those columns: Col(A)=Span{a1,a2,…,an}\text{Col}(A) = \text{Span}\{\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n\}. Equivalently, Col(A)={Ax∣x∈Rn}\text{Col}(A) = \{A\mathbf{x} \mid \mathbf{x} \in \mathbb{R}^n\}.

Key Formula

Col(A)={Ax∣x∈Rn}\text{Col}(A) = \{A\mathbf{x} \mid \mathbf{x} \in \mathbb{R}^n\}
Where:
  • AA = An m × n matrix
  • x\mathbf{x} = Any vector in ℝⁿ
  • Col(A)\text{Col}(A) = A subspace of ℝᵐ

How It Works

To find the column space, row-reduce the matrix to echelon form and identify which columns contain pivots. The corresponding columns of the original matrix (not the reduced matrix) form a basis for the column space. The number of pivot columns equals the rank of the matrix, which is the dimension of the column space. A system Ax=bA\mathbf{x} = \mathbf{b} has a solution if and only if b\mathbf{b} lies in Col(A)\text{Col}(A).

Worked Example

Problem: Find a basis for the column space of the matrix A.
Given matrix: Start with the matrix A.
A=[1232473610]A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 7 \\ 3 & 6 & 10 \end{bmatrix}
Row reduce: Apply row operations: R₂ ← R₂ − 2R₁ and R₃ ← R₃ − 3R₁, then R₃ ← R₃ − R₂.
[123001000]\begin{bmatrix} 1 & 2 & 3 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}
Identify pivot columns: Pivots appear in columns 1 and 3 of the echelon form. Take those columns from the original matrix A.
{[123],  [3710]}\left\{ \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix},\; \begin{bmatrix} 3 \\ 7 \\ 10 \end{bmatrix} \right\}
Answer: A basis for Col(A) is {(1, 2, 3)ᵀ, (3, 7, 10)ᵀ}, and the column space has dimension 2 (rank 2).

Why It Matters

Column space is central to determining whether a linear system Ax=bA\mathbf{x} = \mathbf{b} has a solution. In applications like data fitting and computer graphics, it characterizes the range of transformations a matrix can produce. Understanding it is essential for courses in linear algebra, differential equations, and machine learning.

Common Mistakes

Mistake: Using the columns of the row-reduced matrix as the basis for the column space.
Correction: Row reduction changes the column space. Identify which columns have pivots in the echelon form, then go back and select those same-numbered columns from the original matrix.