Matrix Definition Rank Definition Linear Map Definition Vector Space Definition Exactness Characterized by Vanishing Homology Algebra Characters Are Class Functions Algebra Existence and Uniqueness of the Highest Root Algebra Dense Polynomial Arithmetic over a Polynomially Bounded Effective Field is Polynomial Time Algebra Irreducible Representations of $\operatorname{SO}(4)$ Representation Theory Change of Basis and Conjugacy Representation Theory Complete Reducibility for Finite Groups Representation Theory S-Module Isomorphisms Under Extension of Scalars Algebra Algebra Area