Rank-Nullity Theorem

The Rank-Nullity theorem is one of the most fundamental results in linear algebra. It says that inputs to a matrix split cleanly into two complementary parts.

The theorem

For any m×n matrix A:

where n is the number of columns (the dimension of the input space), rank(A) = dim(column space), and nullity(A) = dim(null space).

What it means

Every input vector splits into two perpendicular components:

• The row space component — this part "survives" the transformation
• The null space component — this part gets annihilated (sent to zero)

Rank counts the surviving dimensions. Nullity counts the lost ones. Together they always add up to the total input dimension n.

Consequences

For a square n×n matrix:

• Rank n → nullity 0 → the matrix is invertible, null space is just zero
• Rank n−1 → nullity 1 → one direction collapses, infinite solutions or no solution
• Rank 0 → everything goes to zero (only possible for the zero matrix)

Try it

Drag the column vectors to change the rank. The bar shows how the two dimensions are split between column space (green) and null space (amber). When vectors are parallel, rank drops to 1 and nullity rises to 1 — they always sum to 2.

import numpy as np
from scipy.linalg import null_space

A = np.array([[1., 2.], [2., 4.]])  # rank 1
n = A.shape[1]  # number of columns = 2

rank = np.linalg.matrix_rank(A)      # 1
nullity = n - rank                    # 1
ns = null_space(A)                    # the null space vector(s)

print(f"rank={rank}, nullity={nullity}, sum={rank+nullity}")  # 1 1 2