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).
Another example is , and the homotopy from the identity transformation to is illustrated above.
When and are finite dimensional, a general linear transformation can be written as a matrix multiplication only after specifying a vector space basis for and . When and have an inner product, and their vector space 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 two-dimensional transformations have a simple classification. Consider the two-dimensional linear transformation