Formally, if V and W are vector spaces over the same ground field K, we say that f : V -> W is a linear transformation if
If V and W are finite dimensional and bases have been chosen, then every linear transformation from V to W can be represented as a matrix; this is useful because it allows concrete calculations. Conversely, matrices yield examples of linear transformations: if A is a real m-by-n matrix, then the rule f(x) = Ax describes a linear transformation Rn -> Rm (see Euclidean space).
There are also important examples of linear transformation involving infinite-dimensional spaces. For instance, the integral yields a linear map from the space of all real-valued integrable functions on the interval [a, b] to R, while differentiation is a linear transformation from the space of all differentiable functions to the space of all functions.
The composition of linear transformations is linear: if f : V -> W and g : W -> Z are linear, then so is g o f : V -> Z. In the finite dimensional case and if bases have been chosen, then the composition of linear maps corresponds to the multiplication of matrices.
If f : V -> W is linear, we define the kernel and the image of f by
Occasionly, V and W can be considered as vector spaces over different ground fields, and it is then important to specify which field was used for the definition of "linear". If V and W are considered as spaces over the field K as above, we talk about K-linear maps. For example, the conjugation of complex numbers is an R-linear map C -> C, but it is not C-linear.