Two subspaces tell most of the story of a linear map. The kernel is everything it sends to zero, which is what the map forgets; the image is everything it reaches. A map loses nothing exactly when its kernel is only the zero vector.

A shadow on a wall forgets how far each point was from the wall, and it covers only part of the wall. The kernel is what is forgotten and the image is the part of the wall that is covered.

The projection $P$ onto the line $y=x$ along the line $y=-x$. Its image is the line $y=x$, and its kernel is the line $y=-x$, which it crushes to $\mathbf{0}$. It sends $\mathbf{v}=(3,1)$ to $P\mathbf{v}=(2,2)$, and the dashed difference $(1,-1)$ lies along the kernel.

The kernel of a linear map $T:V\to W$ is the set of vectors it sends to $\mathbf{0}$, and its image is the set of all its values. The kernel is a subspace of $V$ and the image a subspace of $W$. $$ is one-to-one, or injective, when different vectors have different images, and onto, or surjective, when every vector of is an image.

• $\ker T$ contains $\mathbf{0}$ and is closed under sums and multiples, because $T$ is linear; so is $\operatorname{im}T$, since $T\mathbf{u}+T\mathbf{v}=T(\mathbf{u}+\mathbf{v})$ and .

Everything in $\ker T$ is sent to the single vector $\mathbf{0}$ of $W$, and pulling $\mathbf{0}$ back through $T$ gives the kernel. The image is the part of $W$ that reaches at all; it is the whole of only when is onto.

If $T$ is one-to-one, only $\mathbf{0}$ can share the image $\mathbf{0}$ with $\mathbf{0}$. Conversely, if two vectors have the same image, linearity sends their difference to $\mathbf{0}$; a kernel holding only $\mathbf{0}$ makes the difference zero, so the two vectors were the same.

A vector of the kernel shares its image with $\mathbf{0}$. If $T$ is one-to-one, it is $\mathbf{0}$.

Conversely, suppose two vectors share an image: linearity puts their difference in the kernel.

A kernel holding only $\mathbf{0}$ forces the difference to be $\mathbf{0}$.

So different vectors have different images: $T$ is one-to-one.

For the map of a matrix $A$, the kernel is the solutions of $A\mathbf{x}=\mathbf{0}$ and the image is the span of the columns.

Here $x+y=0$, so the kernel is the line spanned by $(1,-1)$; both columns are $(1,1)$, so the image is the line spanned by .

• $\ker T$: the vectors sent to $\mathbf{0}$, a subspace of the domain.
• $\operatorname{im}T$: the vectors reached, a subspace of the codomain.
• For a matrix: $\ker=\operatorname{Nul}A$ and .

Great work! You now know how to:

• find the kernel and the image of a linear map;
• use $\ker=\operatorname{Nul}A$ and $\operatorname{im}=\operatorname{Col}A$ for a matrix;
• decide one-to-one from the kernel and onto from the image;
• read kernels and images of maps between polynomial spaces.