A function is linear if it maps vectors to vectors, and if for all vectors , and scalers we have

In other word, whether we add the vectors and then apply the map, or apply the map and then add the vectors gives the same result.

linear_maps_preserve_vector_space_properties.jpg

The above property is sometimes called the superposition principle in physics and engineering.

For example, the function is not a linear function, but instead an affine function.

Geometric Intuition

A linear map must have a fixed origin and lines must map into lines.

linear_map_intuition.png

Usefulness

The property of linear transformation has a few advantages:

  • computationally it is much easier to solve system of linear equation than the nonlinear ones
  • Linear transformations can be composed as matrix multiplication
  • All maps can be approximated as linear maps over a short distance/time (Taylor’s theorem)