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

<math>\mathbf{A} = \begin{pmatrix}

-1 & 1 \\ 1 & 0 \end{pmatrix}</math>.

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

<math>\begin{pmatrix}

x \\ y \end{pmatrix}</math>

in the plane, there is another vector in the plane

<math>\begin{pmatrix}

y \\ x+y \end{pmatrix}</math>

such that

<math>\begin{pmatrix} -1 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix}y \\x+y \end{pmatrix} = \begin{pmatrix}x \\y \end{pmatrix}</math>.

Example 2

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

<math>\mathbf{A} = \begin{pmatrix}

3 & 1 \\ -6 & -2 \end{pmatrix}</math>.

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

<math>\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)</math>,

in other words, all points in the plane are carried to the line <math>y=-3x \ </math>.

We write

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