Explore the connections between linear algebra and geometry in this 28-minute video lecture from the "Wild Linear Algebra" series. Delve into how linear algebra provides a robust framework for geometry, replacing Euclid's axiomatic approach with more logically sound definitions and proofs. Examine the algebraic approach to geometry using concepts from Rational Trigonometry and the central role of the dot product. Investigate Euclid's understanding of Pythagoras' theorem and its relationship to perpendicularity. Learn about basic affine geometry and how Euclidean geometry combines affine geometry with metrical structure. Gain insights into the wide-ranging applications of linear algebra in various mathematical fields, including operator algebras, representation theory, harmonic analysis, and algebraic geometry.