Abstract Algebra/Linear Algebra

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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\rightarrow 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\rightarrow W with dimensions n and m and given bases \phi and \psi induces a unique linear map [f]_{\phi}^\psi\,:\,\mathbb{R}^n \rightarrow \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.