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.
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