Book Notes: The Dark Art of Linear Algebra by Seth Braver (ruslanspivak.com)

Seth Braver’s The Dark Art of Linear Algebra (DALA) chapter 1, reviewed here, reintroduces vectors and basic linear-algebra ideas through a geometric, visual lens rather than symbolic manipulation. The chapter treats a vector as an arrow or displacement, demonstrates vector addition (parallelogram/triangle constructions) and subtraction-by-addition, and explains scalar multiplication as stretching, shrinking, or flipping. It then builds to standard basis vectors (i, j, k), vector length via the Pythagorean formula generalized to R^n, and the dot product—connected intuitively to concepts like work—while proving standard properties (commutativity, distributivity, perpendicularity tests) from first principles. The book’s companion videos and exercises reinforce the visuals and derivations, making coordinates feel like descriptions of vectors rather than the objects themselves. For the AI/ML community this matters because geometric intuition underpins embeddings, similarity measures, linear transforms, and optimization. Grasping vectors as geometric objects and the dot product’s role in projections and orthogonality clarifies why common operations—cosine similarity, projections, and linear layers—behave as they do. Practical study tips: watch the short videos, work the exercises, revisit slowly, and use active recall (e.g., Anki). DALA’s approach is recommended for engineers and researchers who want a firmer conceptual foundation for the linear algebra that powers modern ML systems.