Explore the foundations of metrical geometry through a novel, algebraic approach to linear algebra in this 28-minute video lecture. Learn about dot products, their bilinear and symmetric properties, and how they relate to Pythagoras' theorem. Discover how to define perpendicularity and quadrance of vectors using purely algebraic concepts, allowing for work with rational numbers. Follow along as an algebraic proof of Pythagoras' theorem is presented, demonstrating its fundamental importance in linear algebra. Conclude by examining symmetric bilinear forms and their connection to 2x2 matrices, setting the stage for understanding relativistic geometry introduced by Einstein and Minkowski.