Image (mathematics)

In mathematics, the image of a linear transformation is its range: all possible values generated by the transformation. A matrix A, which is an expression of a function, has an image denoted by im(A).

If the rows (or columns, equivalently) of a matrix A are linearly independent, then the image of that transformation is the entire space it is applied to.

Examples

Example 1

Consider all the points (vectors) in the plane, i.e., (x,y) acting under the transformation

$\mathbf {A} ={\begin{pmatrix}-1&1\\1&0\end{pmatrix}}$.

We can be sure that the image of this transformation is the entire plane, because for any point

${\begin{pmatrix}x\\y\end{pmatrix}}$

in the plane, there is another vector in the plane

${\begin{pmatrix}y\\x+y\end{pmatrix}}$

such that

${\begin{pmatrix}-1&1\\1&0\end{pmatrix}}{\begin{pmatrix}y\\x+y\end{pmatrix}}={\begin{pmatrix}x\\y\end{pmatrix}}$.

Example 2

We continue to work in the plane, but now we examine the matrix

$\mathbf {A} ={\begin{pmatrix}3&1\\-6&-2\end{pmatrix}}$.

Now if we examine how this matrix acts on an arbitrary point (a,b), we find that point is carried to

$\mathbf {A} ={\begin{pmatrix}3&1\\-6&-2\end{pmatrix}}{\begin{pmatrix}a\\b\end{pmatrix}}={\begin{pmatrix}3a+b\\-6a-2b\end{pmatrix}}={\begin{pmatrix}1\\-3\end{pmatrix}}(a+b)$,

in other words, all points in the plane are carried to the line $y=-3x\$.

We write

$im(\mathbf {A} )={\begin{pmatrix}1\\-3\end{pmatrix}}t,\forall t\in \mathbb {R} \$.