A linear transformation (also called a linear mapping) is a transformation such that ${\displaystyle {\rm T}:\R^n\to\R^m}$ satisfies the following conditions:

• ${\displaystyle {\rm T}(\vec{a}+\vec{b})={\rm T}(\vec{a})+{\rm T}(\vec{b})}$
• ${\displaystyle {\rm T}(c\vec{a})=c{\rm T}(\vec{a})}$

If a transformation is linear, there will be an associated transformation matrix.

Examples

The transformation

${\displaystyle {\rm T}\begin{bmatrix}x_1\\x_2\end{bmatrix}=\begin{bmatrix}x_2\\3x_2\end{bmatrix}}$

is linear because

${\displaystyle {\rm T}\begin{bmatrix}cx_1\\cx_2\end{bmatrix}=\begin{bmatrix}cx_2\\3cx_2\end{bmatrix} =c\begin{bmatrix}x_2\\3x_2\end{bmatrix}=c{\rm T}\begin{bmatrix}x_1\\x_2\end{bmatrix}}$

and

${\displaystyle {\rm T}\begin{bmatrix}a+b\\c+d\end{bmatrix}=\begin{bmatrix}c+d\\3c+3d\end{bmatrix} =\begin{bmatrix}c\\3c\end{bmatrix}+\begin{bmatrix}d\\3d\end{bmatrix}={\rm T}\begin{bmatrix}a\\c\end{bmatrix}+{\rm T}\begin{bmatrix}b\\d\end{bmatrix}}$

The transformation

${\displaystyle {\rm T}\begin{bmatrix}x_1\\x_2\end{bmatrix}=\begin{bmatrix}x_1x_2\\5\end{bmatrix}}$

is not linear because

${\displaystyle {\rm T}\begin{bmatrix}cx_1\\cx_2\end{bmatrix}=\begin{bmatrix}cx_1cx_2\\5\end{bmatrix} =\begin{bmatrix}c^2x_1x_2\\5\end{bmatrix}\ne c{\rm T}\begin{bmatrix}x_1\\x_2\end{bmatrix}=c\begin{bmatrix}x_1x_2\\5\end{bmatrix}}$

In general, if any variable is raised to a power or two variables are multiplied by the transformation, or if there are any constants other than 0 as elements of the transformed vector, the transformation will not be linear.