Rank-Nullity Theorem — Definition, Formula & Examples

The Rank-Nullity Theorem states that for any matrix, the number of columns equals the rank (dimension of the column space) plus the nullity (dimension of the null space). It tells you exactly how the 'information' in a matrix splits between independent directions and free variables.

Let $A$ be an $m \times n$ matrix. Then $\operatorname{rank}(A) + \operatorname{nullity}(A) = n$ , where $\operatorname{rank}(A) = \dim(\operatorname{Col}(A))$ is the dimension of the column space and $\operatorname{nullity}(A) = \dim(\operatorname{Null}(A))$ is the dimension of the null space (kernel) of $A$ .

Key Formula

\operatorname{rank}(A) + \operatorname{nullity}(A) = n

Where:

$A$ = An m × n matrix
$\operatorname{rank}(A)$ = Number of pivot columns (dimension of the column space)
$\operatorname{nullity}(A)$ = Number of free variables (dimension of the null space)
$n$ = Total number of columns of A

How It Works

To apply the theorem, row-reduce the matrix to echelon form. Count the pivot columns — that number is the rank. The remaining non-pivot columns correspond to free variables, and their count is the nullity. The theorem guarantees these two numbers always sum to

n

, the total number of columns. This means if you know any two of rank, nullity, and

n

, you can immediately find the third.

Worked Example

Problem: Find the rank and nullity of the matrix A, then verify the Rank-Nullity Theorem.

Given matrix: Consider the 2 × 4 matrix:

A = \begin{pmatrix} 1 & 2 & 0 & 1 \\ 0 & 0 & 1 & 3 \end{pmatrix}

Find the rank: The matrix is already in echelon form. There are pivot positions in columns 1 and 3, so:

\operatorname{rank}(A) = 2

Find the nullity: Columns 2 and 4 are non-pivot (free variable) columns, so:

\operatorname{nullity}(A) = 2

Verify the theorem: The matrix has n = 4 columns. Check that rank + nullity = n:

2 + 2 = 4 \checkmark

Answer: The rank is 2, the nullity is 2, and their sum equals the number of columns (4), confirming the Rank-Nullity Theorem.

Why It Matters

The Rank-Nullity Theorem is essential for determining whether a system

A\mathbf{x} = \mathbf{b}

has unique solutions, infinitely many solutions, or none. Engineers use it when analyzing degrees of freedom in mechanical systems, and it appears throughout linear algebra courses when studying invertibility, linear transformations, and eigenspaces.

Common Mistakes

Mistake: Using the number of rows instead of columns on the right side of the equation.

Correction: The theorem says rank + nullity = n, where n is the number of columns, not rows. The number of rows determines the maximum possible rank but does not appear directly in the formula.