This section of the Handbook describes the Magma facilities for linear algebra and module theory. Since this topic is absolutely fundamental for much of algebra, it is important that the reader understand how linear algebra is presented in Magma. The structures covered under this heading include:

• Vector spaces;
• Inner product spaces;
• Modules defined over any ring or algebra
• R[G]-modules, where R is a ring and G is a group;
• Linear transformations and R-module homomorphisms.

In the Magma universe, rectangular matrices are regarded as forming a module (actually a bimodule). We shall regard a rectangular matrix as the concrete realization of a linear transformation or R-module homomorphism. Thus, an m x n matrix over a ring R is considered to be an element of the module Hom_R(M, N). Reflecting the dual nature of matrices, the Hom_R(M, N) operations include the standard module-theoretic operations as well as operations that interpret an element of Hom_R(M, N) as a homomorphism.