[Linear Algebra Series Part 15] Subspaces, Column Space, and Null Space

What this post covers

This post introduces subspace, column space, and null space.

• What a subspace is
• Why column space describes all possible outputs
• Why null space describes the input directions that disappear
• How these spaces explain expressiveness and indistinguishability

Key terms

• subspace: a subset of a vector space that is itself a vector space
• column space: all vectors a matrix can output
• null space: all vectors a matrix sends to zero
• homogeneous equation: a system of the form Ax = 0

Core idea

Once we ask what Ax can produce, two questions naturally appear:

• where can the outputs go?
• which input directions disappear?

These questions are answered by two important subspaces.

A subspace must satisfy three conditions:

1. it contains the zero vector
2. it is closed under vector addition
3. it is closed under scalar multiplication

Many of the most important sets in linear algebra are subspaces, especially column space and null space.

The column space is the set of all possible outputs Ax.

The null space is the set of all inputs that satisfy Ax = 0.

Together, they tell us what a matrix can express and what differences between inputs it cannot detect.

Step-by-step examples

Example 1) Column space

Consider a matrix from R^3 to R^2:

A = [1 0 1
     0 1 1]

Its columns are [1; 0], [0; 1], and [1; 1]. Since the first two already span R^2, this matrix can produce every vector in R^2. So its column space is all of R^2, which here is a subspace of the codomain R^2.

Example 2) Null space

Now solve Ax = 0 for the same matrix. Because there are three unknowns but only two equations, we should expect one free variable.

x1 + x3 = 0
x2 + x3 = 0

If we let x3 = t, then

x = t[-1; -1; 1]

So the null space is a line generated by [-1; -1; 1]. For instance, A[1; 1; -1] = [0; 0], so different nonzero inputs can collapse to the same output.

Example 3) Why equal outputs matter

If two inputs satisfy

Ax1 = Ax2

then

A(x1 - x2) = 0

So their difference lies in the null space. That means the matrix cannot distinguish those directions.

Example 4) Projection and loss of one direction

Projecting 3D data onto the xy-plane keeps two directions and erases the perpendicular one. The output plane is the column space. The lost direction belongs to the null space.

Math notes

• The column space is the span of the matrix columns.
• The null space is the solution set of the homogeneous equation Ax = 0.
• If b belongs to the column space, then Ax = b has a solution.
• If x_particular is one solution of Ax = b, then every solution has the form

x = x_particular + x_null

with x_null in the null space.

Indeed,

A(x_particular + x_null) = Ax_particular + Ax_null = b + 0 = b

so adding a null-space vector does not change the output.

Common mistakes

Thinking a subspace is just any smaller set

It is not. It must contain zero and remain closed under addition and scalar multiplication.

Thinking null space is secondary

Null space is central to freedom, ambiguity, compression, and nonuniqueness.

Memorizing names without meaning

Column space is about possible outputs. Null space is about input directions the map cannot distinguish.

Practice or extension

1. What happens if b lies outside the column space?
2. What does a larger null space tell you about information loss?
3. Why is the solution set of Ax = 0 a subspace?

Wrap-up

This post introduced three key spaces.

• Column space describes what a matrix can produce.
• Null space describes what a matrix erases.
• Together they explain both expressive power and ambiguity.
• Next, we summarize those ideas numerically using rank and nullity.