Kernel of a linear map

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 kernel of $T$ is $$\ker T = \{v \in V : T(v) = 0\}.$$

Remarks

The kernel is a linear subspace of $V$. A linear map is injective if and only if its kernel is trivial. For the map $x \mapsto Ax$ of a matrix $A$, the kernel is the null space of $A$, the solution set of $Ax = 0$. The kernel of the projection $\R^3 \to \R^2$, $(x, y, z) \mapsto (x, y)$, is the $z$-axis.

Explanations

L1 human

The kernel collects the vectors that the map sends to zero. It measures how much information the map loses. If only the zero vector goes to zero, then the map loses nothing.