How to Add Geometry to A Vector Space (Inner Product Space)

In this video, we explore how norms, metrics, and inner products give geometric meaning to vector spaces. Starting from the basic structure of a vector space, we show why addition and scalar multiplication alone are not enough to describe concepts like length, distance, and angle, and how introducing an inner product allows us to build the geometry of an inner product space. We then investigate the relationship between norms, metrics, and inner products, comparing the Euclidean norm and Manhattan norm, and showing how different choices of norm change the geometry of the space. You’ll also see how symmetric positive definite (SPD) matrices define new inner products and induce new notions of distance and angle, transforming circles into ellipses and altering the geometry of the vector space itself. This video builds the mathematical intuition behind metric spaces, normed spaces, orthogonality, and generalized Euclidean geometry in linear algebra. 00:00 Inner Product Space 03:08 Norm (Manhattan and Euclidean norm) 07:13 Distance and Metric 13:11 Angle and Orthogonality