  This set of notes re-proves known results on weighted automata (over a field,
also known as multiplicity automata). The text offers a unified view on
theorems and proofs that have appeared in the literature over decades and were
written in different styles and contexts. None of the results reported here are
claimed to be new. The content centres around fundamentals of equivalence and
minimization, with an emphasis on algorithmic aspects. The presentation is
minimalistic. No attempt has been made to motivate the material. Weighted
automata are viewed from a linear-algebra angle. As a consequence, the proofs,
which are meant to be succinct, but complete and almost self-contained, rely
mainly on elementary linear algebra.
