A linear transformation between two vector spaces and is a map such that the following hold:
1. for any vectors and in , and
2. for any scalar .
A linear transformation may or may not be injective or surjective. When and have the same dimension, it is possible for to be invertible, meaning there exists a such that . It is always the case that . Also, a linear transformation always maps lines to lines (or to zero).


The main example of a linear transformation is given by matrix multiplication. Given an matrix , define , where is written as a column vector (with coordinates). For example, consider
(1)

then is a linear transformation from to , defined by
(2)

When and are finite dimensional, a general linear transformation can be written as a matrix multiplication only after specifying a vector basis for and . When and have an inner product, and their vector bases, and , are orthonormal, it is easy to write the corresponding matrix . In particular, . Note that when using the standard basis for and , the th column corresponds to the image of the th standard basis vector.
When and are infinite dimensional, then it is possible for a linear transformation to not be continuous. For example, let be the space of polynomials in one variable, and be the derivative. Then , which is not continuous because while does not converge.
Linear twodimensional transformations have a simple classification. Consider the twodimensional linear transformation
(3)
 
(4)

Now rescale by defining and . Then the above equations become
(5)

where and , , , and are defined in terms of the old constants. Solving for gives
(6)

so the transformation is onetoone. To find the fixed points of the transformation, set to obtain
(7)

This gives two fixed points, which may be distinct or coincident. The fixed points are classified as follows.