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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.

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