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 and with bases and and dimensions and , respectively, a linear map corresponds to a unique 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 , 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 is isomorphic to for some field and some , called the dimension of . Specifying such an isomorphism is equivalent to choosing a basis for . Thus, any linear map between vector spaces with dimensions and and given bases and induces a unique linear map . These maps are presicely the matrices, and the matrix in question is called the matrix representation of relative to the bases .

Remark: The idea of identifying a basis of a vector space with an isomorphism to may be new to the reader, but the basic principle is the same.