Linear Equations and Vector Spaces

This lecture covers the solutions of homogeneous linear equations in matrix form, defining the null space of a matrix as a vector subspace of R^n. It also discusses the properties of subspaces, including the closure under addition and scalar multiplication. The lecture introduces the concept of vector spaces, defining the ten axioms that must be satisfied for a set to be considered a vector space. It explores the linear independence of vectors and the generation of subspaces through linear combinations. Additionally, it delves into bases, dimensions, and the distinction between linearly independent and dependent vectors.