According to Wikipedia ("Linear algebra", 03-Mar-2019,
https://en.wikipedia.org/wiki/Linear_algebra) "Linear algebra is the
branch of mathematics concerning linear equations [...], linear functions
[...] and their representations through matrices and vector spaces." Or
according to the Merriam-Webster dictionary ("linear algebra", 12-Mar-2019,
https://www.merriam-webster.com/dictionary/linear%20algebra) "Definition
of linear algebra: a branch of mathematics that is concerned with
mathematical structures closed under the operations of addition and scalar
multiplication and that includes the theory of systems of linear equations,
matrices, determinants, vector spaces, and linear transformations." Dealing
with modules (over rings) instead of vector spaces (over fields) allows for a
more unified approach. Therefore, linear equations, matrices, determinants,
are usually regarded as "over a ring" in this part.

Unless otherwise stated, the rings of scalars need not be commutative
(see df-cring), but the existence of a multiplicative neutral element is
always assumed (our rings are unital, see df-ring).

For readers knowing vector spaces but unfamiliar with modules: the elements
of a module are still called "vectors" and they still form a group under
addition, with a zero vector as neutral element, like in a vector space.
Like in a vector space, vectors can be multiplied by scalars, with the usual
rules, the only difference being that the scalars are only required to form a
ring, and not necessarily a field or a division ring. Note that any vector
space is a (special kind of) module, so any theorem proved below for modules
applies to any vector space.