A Linear Algebraic Approach to Quaternions

Unit quaternions are a powerful way to represent rotations within computer graphics and physics applications. Unfortunately, the mathematical complexity of quaternions seems to discourage some practitioners from any attempts at understanding them. This document provides an alternate approach to the presentation of quaternions, one that is based solely on concepts from trigonometry and linear algebra. The algebra and geometry of quaternions is motivated from a study of certain rotation matrices in four dimensions. This is in contrast to the classical approach that defines unit quaternions as points on a unit hypersphere in four dimensions and lists their important algebraic properties to be taken on faith.