1 Preliminaries

Linear algebra is written in the language of sets, relations and functions, and almost every later definition is phrased in it. This short section fixes that language.

Three ideas earn their place. Sets and the operations on them supply the notation used throughout. Equivalence relations are the tool for regarding different objects as the same for a given purpose, and the theorem that their classes partition a set is the reason that idea is safe; the same argument reappears whenever matrices are classified up to equivalence or similarity. Functions, finally, are what linear transformations are: Section 6 studies functions between vector spaces that respect addition and scaling, so the distinctions drawn here — injective, surjective, inverse image — are used there without further comment.

A reader confident with this material can move to Section 2 and refer back.

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