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The four fundamental subspaces, side by side

Row space, column space, null space, and left null space — where each lives, its dimension, and how they pair up.

Any question of the form "what space does this vector live in, and what's its dimension" is answered by matching it to one of these four. Row space and null space live in R^n (the input side); column space and left null space live in R^m (the output side, for an m x n matrix). Rank ties row space and column space together; nullity ties null space to the leftover free variables.

The four spaces

Term

row space

See section →
\operatorname{Row}(A) — the span of a matrix's rows.

Spanned by A's rows — lives in R^n.

See section →
\operatorname{Col}(A) — the span of a matrix's columns.

Spanned by A's columns — lives in R^m.

Term

null space

See section →
\operatorname{Nul}(A) — every \mathbf x with A\mathbf x = \mathbf 0.

All solutions to Ax = 0 — lives in R^n.

See section →
\operatorname{Nul}(A^{\mathsf T}) — vectors \mathbf y with \mathbf y^{\mathsf T}A=\mathbf 0.

The null space of A transpose — lives in R^m.

How to find a basis for each from one RREF

Dimensions and how they relate

Term

rank

See section →
\operatorname{rank}(A) — the shared dimension of the row space and column space.

The shared dimension of row space and column space.

Term

nullity

See section →
\operatorname{nullity}(A) = \dim\operatorname{Nul}(A), the number of free variables.

The dimension of the null space.

How the pairs sit orthogonally

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