Introduction

The column space (also called the image) of a matrix is the set of all vectors you can get by multiplying that matrix by some input vector.
If you already understand span, then the column space will feel familiar: it is simply the span of the columns of the matrix.

Review: Computing Matrix–Vector Products

• Let's quickly review what it means to multiply a vector by a matrix: $$\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} ax_1 + bx_2 \\ cx_1 + dx_2 \end{bmatrix} = \begin{bmatrix} ax_1 \\ cx_1 \end{bmatrix} + \begin{bmatrix} bx_2 \\ dx_2 \end{bmatrix}$$
• The result is simply the sum of the columns in the matrix, multiplied by it's corresponding element in the vector.

What the Column Space Represents

• A matrix $A$ acts like a machine: it takes an input vector $x$ and produces an output $Ax$.
• The column space is the collection of all possible outputs the machine can produce.
• If $A$ has columns $a_1, a_2, \dots, a_n$, then every output looks like $$Ax = x_1 a_1 + x_2 a_2 + \dots + x_n a_n.$$
• So the column space is $$\text{Col}(A) = \text{Span}\{a_1, a_2, \dots, a_n\}.$$

Why the Column Space Matters

• It tells you which vectors are reachable using the matrix.
• It helps answer questions like:
  • Does the system $Ax = b$ have a solution?
  • What directions can the matrix “move” vectors into?
  • Does the matrix cover the whole space or only part of it?

Examples

Example 1: A 2×2 matrix

Let $$A = \begin{bmatrix}1 & 2 \\ 2 & 4\end{bmatrix}.$$

• Columns: $(1,2)$ and $(2,4)$.
• Notice that $(2,4) = 2(1,2)$.
• So the columns lie on the same line.
• Therefore the column space is the line spanned by $(1,2)$.

Example 2: A 3×2 matrix

Let $$A = \begin{bmatrix}1 & 0 \\ 0 & 1 \\ 1 & 1\end{bmatrix}.$$

• Columns: $(1,0,1)$ and $(0,1,1)$.
• These two vectors are not multiples of each other.
• Their span is a plane inside $\mathbb{R}^3$.
• So the column space is a 2‑dimensional plane.

How to Determine the Column Space

To find the column space:

• Identify the columns of the matrix.
• Check whether some columns are linear combinations of others.
• Keep only the pivot columns after row‑reducing the matrix.
• The column space is the span of those pivot columns (from the original matrix).

This method avoids unnecessary work and reveals the dimension of the column space.

Geometric Interpretation

• In $\mathbb{R}^2$, the column space is either:
  • a point (only if the matrix is zero),
  • a line,
  • or the entire plane.
• In $\mathbb{R}^3$, it can be:
  • a point,
  • a line,
  • a plane,
  • or all of $\mathbb{R}^3$.

The dimension of the column space is the rank of the matrix.

Exercises

1. Find the column space of $$A = \begin{bmatrix}1 & 3 \\ 2 & 6\end{bmatrix}.$$

   Solution

   Columns: $(1,2)$ and $(3,6) = 3(1,2)$.
   Column space is the line spanned by $(1,2)$.
2. Determine whether the vector $(3,5)$ is in the column space of $$A = \begin{bmatrix}1 & 1 \\ 2 & 3\end{bmatrix}.$$

   Solution

   Solve $x(1,2) + y(1,3) = (3,5)$.
   A solution exists: $x = 4$, $y = -1$.
   So $(3,5)$ is in the column space.
3. Describe the column space of $$A = \begin{bmatrix}1 & 0 & 1 \\ 0 & 1 & 1\end{bmatrix}.$$

   Solution

   Columns: $(1,0)$, $(0,1)$, $(1,1)$.
   First two already span all of $\mathbb{R}^2$.
   Column space is $\mathbb{R}^2$.
4. True or false: The column space of a $3 \times 2$ matrix can be all of $\mathbb{R}^3$.

   Solution

   False.
   A $3 \times 2$ matrix has at most rank 2, so its column space cannot fill $\mathbb{R}^3$.
5. Find a basis for the column space of $$A = \begin{bmatrix}2 & 4 & 1 \\ 1 & 2 & 0 \\ 3 & 6 & 1\end{bmatrix}.$$

   Solution

   Row‑reduce to find pivot columns.
   Pivot columns are the first and third.
   Basis: $\{(2,1,3), (1,0,1)\}$.
6. Determine the dimension of the column space of $$A = \begin{bmatrix}1 & 2 & 3 \\ 0 & 1 & 4 \\ 0 & 0 & 0\end{bmatrix}.$$

   Solution

   Pivot columns are the first two.
   Dimension of the column space is 2.
7. Explain in words what it means for a vector $b$ to not be in the column space of a matrix $A$.

   Solution

   It means there is no combination of the columns of $A$ that equals $b$.
   In other words, the system $Ax = b$ has no solution.