[Linear Algebra Series Part 16] Rank and Nullity: How Much Information Survives and How Much Disappears

What this post covers

This post introduces rank and nullity.

• What rank counts
• What nullity counts
• How they summarize preserved versus lost directions
• Why the rank-nullity theorem matters

Key terms

• rank: the dimension of the column space
• nullity: the dimension of the null space
• pivot: a leading nonzero position in echelon form
• free variable: a variable from a non-pivot column

Core idea

After learning column space and null space, the next step is to measure their sizes.

That is what rank and nullity do.

Rank is the number of independent output directions a matrix can produce. Equivalently, it is the dimension of the column space.

Nullity is the number of independent input directions that disappear. Equivalently, it is the dimension of the null space.

Together, they tell us how the input space splits into a part that survives and a part that is lost. This count is about independent directions, not just how many rows or columns happen to be written down.

Step-by-step examples

Example 1) A simple 2 x 3 matrix

Consider

A = [1 0 0
     0 1 0]

This map sends [x; y; z] to [x; y]. The x and y directions survive, but the z direction disappears.

So:

• rank(A) = 2
• nullity(A) = 1

To see the nullity directly, solve Ax = 0. We get x = 0, y = 0, while z is free, so the null space is all multiples of [0; 0; 1] and has dimension 1.

Example 2) Projection onto a plane

The matrix

P = [1 0 0
     0 1 0
     0 0 0]

projects 3D vectors onto the xy-plane. The output plane has dimension 2, so the rank is 2. The lost z direction has dimension 1, so the nullity is 1.

Example 3) Repeated columns

If

A = [1 2
     2 4]

then the second column is a multiple of the first. So the matrix has two columns but only one independent output direction. Its rank is 1.

Math notes

• Rank equals the number of pivots, and each pivot corresponds to one independent column direction.
• Nullity equals the number of free variables in Ax = 0.
• Row rank and column rank are always equal.
• If A is an m x n matrix, where n is the number of columns, then

n = rank(A) + nullity(A)

This is the rank-nullity theorem.

A square matrix is invertible if and only if it has full rank. If rank is smaller than the input dimension, some information must be lost.

Common mistakes

Memorizing rank as only a row-count trick

Rank is really about independent output directions, not just a mechanical count.

Treating nullity as secondary

Nullity tells you how much ambiguity or compression remains in the input space.

Thinking size alone determines rank

The matrix shape gives an upper bound, but dependence can make the actual rank smaller.

Practice or extension

1. If a 3 x 3 matrix has rank 2, what must be true about its nullity?
2. If the input dimension is 5 and the rank is 3, what is the nullity?
3. Why can a square matrix be noninvertible even when it looks large enough?

Wrap-up

This post introduced rank and nullity.

• Rank counts independent output directions.
• Nullity counts input directions that vanish.
• Their sum recovers the input dimension.
• Next, we use orthogonality and projection to build the idea of best approximation.