[<< wikibooks] Abstract Algebra/Linear Algebra
The reader is expected to have some familiarity with linear algebra. For example, statements such as

Given vector spaces

V

{\displaystyle V}
and

W

{\displaystyle W}
with bases

B

{\displaystyle B}
and

C

{\displaystyle C}
and dimensions

n

{\displaystyle n}
and

m

{\displaystyle m}
, respectively, a linear map

f

:

V
→
W

{\displaystyle f\,:\,V\to W}
corresponds to a unique

m
×
n

{\displaystyle m\times n}
matrix, dependent on the particular choice of basis.should be familiar. It is impossible to give a summary of the relevant topics of linear algebra in one section, so the reader is advised to take a look at the linear algebra book.
In any case, the core of linear algebra is the study of linear functions, that is, functions with the property

f
(
α
x
+
β
y
)
=
α
f
(
x
)
+
β
f
(
y
)

{\displaystyle f(\alpha x+\beta y)=\alpha f(x)+\beta f(y)}
, where greek letters are scalars and roman letters are vectors.
The core of the theory of finitely generated vector spaces is the following:
Every finite-dimensional vector space

V

{\displaystyle V}
is isomorphic to

F

n

{\displaystyle \mathbb {F} ^{n}}
for some field

F

{\displaystyle \mathbb {F} }
and some

n
∈

N

{\displaystyle n\in \mathbb {N} }
, called the dimension of

V

{\displaystyle V}
. Specifying such an isomorphism is equivalent to choosing a basis for

V

{\displaystyle V}
. Thus, any linear map between vector spaces

f

:

V
→
W

{\displaystyle f\,:\,V\to W}
with dimensions

n

{\displaystyle n}
and

m

{\displaystyle m}
and given bases

ϕ

{\displaystyle \phi }
and

ψ

{\displaystyle \psi }
induces a unique linear map

[
f

]

ϕ

ψ

:

R

n

→

R

m

{\displaystyle [f]_{\phi }^{\psi }\,:\,\mathbb {R} ^{n}\to \mathbb {R} ^{m}}
. These maps are presicely the

m
×
n

{\displaystyle m\times n}
matrices, and the matrix in question is called the matrix representation of

f

{\displaystyle f}
relative to the bases

ϕ
,
ψ

{\displaystyle \phi ,\psi }
.
Remark: The idea of identifying a basis of a vector space with an isomorphism to

F

n

{\displaystyle \mathbb {F} ^{n}}
may be new to the reader, but the basic principle is the same.