Abstract Algebra/Linear Algebra

The reader is expected to have some familiarity with linear algebra. For example, statements such as

Given vector spaces

V

and

W

with bases

B

and

C

and dimensions

n

and

m

, respectively, a linear map

f\,:\,V\to W

corresponds to a unique

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 $f(\alpha x+\beta y)=\alpha f(x)+\beta f(y)$ , 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 $V$ is isomorphic to $\mathbb {F} ^{n}$ for some field $\mathbb {F}$ and some $n\in \mathbb {N}$ , called the dimension of $V$ . Specifying such an isomorphism is equivalent to choosing a basis for $V$ . Thus, any linear map between vector spaces $f\,:\,V\to W$ with dimensions $n$ and $m$ and given bases $\phi$ and $\psi$ induces a unique linear map $[f]_{\phi }^{\psi }\,:\,\mathbb {R} ^{n}\to \mathbb {R} ^{m}$ . These maps are presicely the $m\times n$ matrices, and the matrix in question is called the matrix representation of $f$ relative to the bases $\phi ,\psi$ .

Remark: The idea of identifying a basis of a vector space with an isomorphism to $\mathbb {F} ^{n}$ may be new to the reader, but the basic principle is the same.

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