A linear transformation can lose some input directions and may fail to reach some possible outputs. The kernel and image describe these two effects.

For

$$T:V\to W,$$

the kernel is

$$\ker T={\mathbf v\in V:T(\mathbf v)=\mathbf0},$$

and the image is

$$\operatorname{im}T={T(\mathbf v):\mathbf v\in V}.$$

Kernel: what is sent to zero

If $T$ is represented by a matrix $A$, then the kernel consists of the solutions of

$$A\mathbf x=\mathbf0.$$

For example,

$$A=\begin{pmatrix}1&1\end{pmatrix}$$

maps $(x,y)$ to $x+y$. Its kernel satisfies

$$x+y=0,$$

so

$$\ker T=\operatorname{span}((1,-1)).$$

Different inputs separated by a kernel vector produce the same output.

Image: what can be reached

For a matrix transformation, the image is the span of the matrix columns. If

$$A=\begin{pmatrix}1&2\2&4\end{pmatrix},$$

both columns lie on the same line, so the image is one-dimensional even though the output space is $\mathbb R^2$.

Both are subspaces

The kernel is a subspace of the domain and the image is a subspace of the codomain. Linearity guarantees closure because linear combinations of zero-producing inputs still produce zero, and linear combinations of reachable outputs are still reachable.

The kernel measures directions lost by the transformation; the image measures the outputs that survive.