Linear algebra

Null space

The null space of A is the set of vectors mapped to zero by A. It is a subspace of the input space.

Ask the Linear algebra assistant1 min read · Updated September 9, 2026

Definition

The null space of A is the set of vectors mapped to zero by A. It is a subspace of the input space.

Intuition

Null-space directions are information that the transformation loses.

Worked example

For A = [1 1], every vector (t, −t) maps to zero, so the null space is a line.

The math

ker⁡(A)={x:Ax=0}\ker(A)=\{x:Ax=0\}. Rank-nullity gives rank⁡(A)+nullity⁡(A)=n\operatorname{rank}(A)+\operatorname{nullity}(A)=n for n columns.

In machine learning

Null-space analysis reveals nonunique solutions and unidentifiable parameter directions.

Sources

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