# Abstract Algebra/Linear Algebra

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

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

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