...the set of matrices is an isomorphic representation of the set of linear functions on a finite dimensional linear space...other than being a mere array of numbers.
These linear functions are defined on a fixed basis, while the matrices are representations of when the functions are applied to arbitrary vectors constructed from the basis vectors. The matrices cannot be defined directly as functions since they are just lists of numbers without variables, i.e. they are just the 'constants' in the functions.
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