Image, preimage, and kernel of a linear transformation

A transformation connects an input with its output. Sometimes we know the input and want to find where it goes. Other times we know an output and want to find every input that produces it. Image, preimage, and kernel give names to those questions.

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Find an image by applying the transformation

If $T\left(x\right)=Ax$, the image of an input $x$ is $Ax$. The image of a set is all the outputs we get by applying $T$ to its members.

Find a preimage by solving for the inputs

$A=\left[\begin{array}{rr}-3& 1\\ 2& 0\end{array}\right]$

Find the preimage of the two-point set containing $(0,0)$ and $(2,3)$. First solve $Ax=(0,0)$, which gives $x=(0,0)$. Then solve $Ax=(2,3)$.

$\begin{array}{l}-3x_1+x_2=2\\ 2x_1=3\\ x_1=\frac{3}{2}\\ x_2=\frac{13}{2}\end{array}$

The full preimage is the two-point set containing $(0,0)$ and $(\frac{3}{2},\frac{13}{2})$.

Find the kernel by using zero as the target

The kernel is the set of inputs satisfying $T\left(x\right)=0$. For a matrix transformation, this is exactly the null space of $A$. For the matrix above, the kernel contains only $(0,0)$.

Preimages can contain several inputs. For example, if $S(x,y)=(x,0)$, the preimage of $(2,0)$ is every point $(2,t)$. Its kernel is every point $(0,t)$. Here $t$ can be any real number.

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