A linear map is a type of vector function which assigns to each vector in one space to a vector in another space. The space which is used as the input to the function is called the domain of the map, the output space is called the codomain.

For the function To be a linear map from a vector space to a vector space (written as ) for any two vectors and in and a constant scalar the following must hold true:

1. Additivity:
2. Homogeneity of degree 1:

Additionally must be defined for all values in .

Linear maps are typically represented using matrices and their application is usually expressed as a matrix product with the input matrix. For example we could define a linear map from as

Then we can apply this map to any vector in as . which gives us the resulting vector in .