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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:
- (i)
- Vector spaces;
- (ii)
- Inner product spaces;
- (iii)
- Modules defined over any ring or algebra
- (iv)
- R[G]-modules, where R is a ring and G is a group;
- (v)
- Linear transformations and R-module homomorphisms.
Although vector spaces are, of course, subsumed under general modules,
we present a separate treatment of them, firstly because of their importance
and secondly because their theory is somewhat cleaner than that of a general
module. Magma users who are unfamiliar with the language of module theory
will find a self-contained treatment of the vector space machinery in
Chapter VECTOR SPACES.
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 HomR(M, N). Reflecting the dual nature of matrices, the
HomR(M, N) operations include the standard module-theoretic operations
as well as operations that interpret an element of HomR(M, N) as a
homomorphism.
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