Rank & Nullity

Discover how the dimension of the column space and the dimension of the null space of a matrix are linked by a single elegant theorem.

Linear Algebra

What Is the Rank of a Matrix?

The rank of a matrix \(A\) is the number of linearly independent columns (or equivalently, rows). It tells you the dimension of the column space \(\text{Col}(A)\).

Full rank

\(\text{rank}(A) = \min(m,n)\): every column (or row) is independent.

Rank-deficient

\(\text{rank}(A) < \min(m,n)\): some columns are redundant.

What Is the Nullity of a Matrix?

The nullity of a matrix \(A\) is the dimension of its null space \(\text{Nul}(A)\): the set of all vectors \(\mathbf{x}\) such that \(A\mathbf{x} = \mathbf{0}\).

Nullity = 0

Only \(\mathbf{x} = \mathbf{0}\) solves \(A\mathbf{x} = \mathbf{0}\): the columns are independent.

Nullity > 0

Infinitely many solutions exist: some directions are "lost" in the mapping.

Visualizing Rank and Nullity

The Rank-Nullity Theorem

For any \(m \times n\) matrix \(A\):

Here \(n\) is the number of columns of \(A\). Every column is either a pivot column (contributing to rank) or leads to a free variable (contributing to nullity).

Finding Rank via Row Reduction

To find the rank, row-reduce \(A\) to row echelon form (REF). The number of non-zero rows (equivalently, pivot positions) equals the rank.

Rank vs Nullity: Key Distinction

	Rank	Nullity
Measures	Dimension of Col(A)	Dimension of Nul(A)
Counts	Pivot columns	Free variables
Meaning	Independent outputs	Directions mapped to \(\mathbf{0}\)
If maximal	If \(\text{rank}(A)=n\), any solution to \(A\mathbf{x}=\mathbf{b}\) is unique.	Every column is free (\(\text{rank}(A) = 0\))
If zero	\(A = \mathbf{0}\) (zero matrix)	Columns are independent

How to Find Rank and Nullity

1

Write the matrix \(A\) and note \(n\), the number of columns.

2

Row-reduce to REF (or RREF). Use elementary row operations: swap, scale, add multiples.

3

Count pivot positions. That count is \(\text{rank}(A)\).

4

Compute nullity: \(\text{nullity}(A) = n - \text{rank}(A)\).

5

Verify: check that \(\text{rank} + \text{nullity} = n\).

Example 1: A 2 × 3 Matrix

Find the rank and nullity of \(A = \begin{bmatrix} 1 & 3 & 5 \\ 2 & 6 & 10 \end{bmatrix}\).

Example 2: A 3 × 4 Matrix

Find the rank and nullity of \(B = \begin{bmatrix} 1 & 0 & 2 & 1 \\ 0 & 1 & -1 & 3 \\ 2 & 1 & 3 & 5 \end{bmatrix}\).

Example 3: Full-Rank Square Matrix

Find the rank and nullity of \(C = \begin{bmatrix} 1 & 2 \\ 3 & 7 \end{bmatrix}\).

What Rank & Nullity Tell Us About Solutions

Consider the system \(A\mathbf{x} = \mathbf{b}\) where \(A\) is \(m \times n\):

Nullity = 0

At most one solution for each \(\mathbf{b}\). If a solution exists, it is unique.

Nullity > 0

If a solution exists, there are infinitely many (a whole affine subspace of dimension = nullity).

Rank, Nullity, and Invertibility

For an \(n \times n\) square matrix \(A\), these are all equivalent:

1

\(A\) is invertible (\(A^{-1}\) exists).

2

\(\text{rank}(A) = n\) (full rank).

3

\(\text{nullity}(A) = 0\) (trivial null space).

4

\(\det(A) \neq 0\).

5

\(A\mathbf{x} = \mathbf{b}\) has a unique solution for every \(\mathbf{b}\).

Rank of \(A\) equals Rank of \(A^T\)

A remarkable fact: the row rank and column rank of any matrix are always equal. This means:

The number of linearly independent rows always equals the number of linearly independent columns. However, nullity can differ:

\(A\) is \(m \times n\)

\(\text{nullity}(A) = n - \text{rank}(A)\)

\(A^T\) is \(n \times m\)

\(\text{nullity}(A^T) = m - \text{rank}(A)\)

Common Mistakes

✕ Using rows instead of columns

Rank-nullity uses \(n\) = number of columns, not rows. Writing \(\text{rank} + \text{nullity} = m\) is wrong.

✕ Counting zero rows as pivots

Only non-zero rows in REF have pivots. A row of all zeros does not contribute to rank.

✕ Forgetting to fully reduce

Stopping too early can hide dependencies. Always reduce until you clearly see all pivot positions.

Summary of Key Facts

Fact	Formula / Rule
Rank-Nullity Theorem	\(\text{rank}(A) + \text{nullity}(A) = n\)
Rank	Number of pivots in REF
Nullity	Number of free variables = \(n - \text{rank}\)
Invertible \(n \times n\)	\(\text{rank}(A) = n\), \(\text{nullity}(A) = 0\)
Row rank = Column rank	\(\text{rank}(A) = \text{rank}(A^T)\)
Unique solution	\(\text{nullity}(A) = 0\)
Infinite solutions	\(\text{nullity}(A) > 0\) (and consistent)

Question 1 of 5

If \(A\) is a \(3 \times 5\) matrix with \(\text{rank}(A) = 3\), what is \(\text{nullity}(A)\)?

By the Rank-Nullity Theorem, \(\text{nullity}(A) = n - \text{rank}(A) = 5 - 3 = 2\). The number of columns is \(n = 5\), not \(m = 3\).

Question 2 of 5

A \(4 \times 4\) matrix \(B\) has \(\text{nullity}(B) = 0\). Which statement is true?

\(\text{nullity}(B) = 0\) means \(\text{rank}(B) = 4 - 0 = 4\) (full rank). A square full-rank matrix is invertible, and its determinant is non-zero.

Question 3 of 5

After row-reducing a \(5 \times 6\) matrix, you find 3 pivot positions. How many free variables are there?

Free variables = \(n - \text{rank} = 6 - 3 = 3\). The number of columns is \(n = 6\), and 3 of them are pivot columns, leaving 3 free.

Question 4 of 5

If \(A\) is a \(2 \times 4\) matrix, what is the maximum possible rank of \(A\)?

The rank can be at most \(\min(m, n) = \min(2, 4) = 2\). You cannot have more pivot positions than the number of rows.

Question 5 of 5

The Rank-Nullity Theorem states that \(\text{rank}(A) + \text{nullity}(A)\) equals which quantity?

The theorem states \(\text{rank}(A) + \text{nullity}(A) = n\), the number of columns. This is because each column corresponds to either a pivot (rank) or a free variable (nullity).