Kernel¶

Definition¶

Let $\boldsymbol{A}\in\mathcal{L}(V,W)$ be a linear map. The kernel or null space of $\boldsymbol{A}$ is $\ker \boldsymbol{A}=\{\boldsymbol{v}\in V\colon\boldsymbol{A}\boldsymbol{v}=\boldsymbol{0}\}$ $0}$ .

Interpretation¶

The kernel contains all input vectors that the linear map sends to the zero vector. It measures what the map loses: two inputs have the same image exactly when their difference lies in the kernel, since $\boldsymbol{A}\boldsymbol{u}=\boldsymbol{A}\boldsymbol{v}$ if and only if $\boldsymbol{A}(\boldsymbol{u}-\boldsymbol{v})=\boldsymbol{0}$ $0$ .

The projection onto the x1-axis: a vector v on the x2-axis, which forms the kernel, is sent to 0, while another vector w is flattened onto the horizontal axis.

The matrix $\boldsymbol{A}=\begin{pmatrix}1&0\\0&0\end{pmatrix}$ flattens the plane onto the $x_1$ -axis: $\boldsymbol{w}$ goes straight down to $\boldsymbol{A}\boldsymbol{w}$ , and every vector on the $x_2$ -axis, such as $\boldsymbol{v}$ , goes to $\boldsymbol{0}$ . So $\ker \boldsymbol{A}=\{\boldsymbol{v}\colon\boldsymbol{A}\boldsymbol{v}=\boldsymbol{0}\}=\operatorname{span}(\boldsymbol{e}_2)$ $span (e_{2})$ , the whole $x_2$ -axis.

Drag $\boldsymbol{x}$ onto the dashed line: there $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{0}$ $0$ . The kernel is $\{\boldsymbol{0}\}$ only when $\det(\boldsymbol{A})\neq0$ $0$ .

For a matrix, finding the kernel means solving the homogeneous system $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{0}$ by row reduction. Each free variable contributes one basis vector, so the dimension of the kernel is the number of free variables. The kernel also describes every other solution set: if $\boldsymbol{A}\boldsymbol{x}_p=\boldsymbol{b}$ $b$ , the solutions of $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ are exactly $\boldsymbol{x}_p$ plus a vector of the kernel.

In particular, a linear map is injective if and only if its kernel is $\{\boldsymbol{0}\}$ , and the fundamental theorem of linear maps links the dimension of the kernel to that of the range.

A common slip is to conclude "the kernel is $\{\boldsymbol{0}\}$ " after checking only that $\boldsymbol{A}\boldsymbol{0}=\boldsymbol{0}$ $0$ . The zero vector lies in every kernel; the question is whether any other vector does. Note also that the kernel is a subspace of the domain $V$ , not of $W$ .

Key Results¶

The kernel of a linear map is a subspace of the domain.
A linear map is injective exactly when its kernel is $\{\boldsymbol{0}\}$ .

Examples¶

For $\boldsymbol{A}(x,y)=(x,0)$ $(x, 0)$ , the kernel is $\{(0,y)\colon y\in\mathbb{R}\}$ $R}$ .
The kernel of the identity map is $\{\boldsymbol{0}\}$ .

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 4
Math V: Advanced Linear Algebra · Week 5 (revisited)

Aliases: null space, nullspace Symbols: element of, real numbers

Last updated: 20 October 2026 03:02