Nullity

Definition

Let $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $T \colon V \to W$ be a linear map. The nullity of $T$ is $$\operatorname{nullity} T = \dim \ker T,$$ the dimension of the kernel of $T$. The nullity of a matrix $A$ is the nullity of the map $x \mapsto Ax$, the dimension of the null space of $A$.

Remarks

The nullity is $0$ if and only if $T$ is injective. For an $m \times n$ matrix, the nullity is the number of free variables of $Ax = 0$. By the rank–nullity theorem, $\rank T + \operatorname{nullity} T = \dim V$ when $V$ is finite-dimensional.

Explanations

L1 human

The nullity counts the dimensions that the map crushes to zero. The projection of space onto a plane crushes one direction, so its nullity is $1$. Rank counts what survives, and nullity counts what is lost.