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.