A linear transformation (also called a linear mapping) is a transformation such that satisfies the following conditions:

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

## Examples

The transformation

is linear because

and

The transformation

is not linear because

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.