Vector Spaces: Properties and Operations

This lecture covers the properties of vector spaces, including the existence of zero and opposite elements, vector addition, and scalar multiplication. It also explores examples of vector addition and functions defined within vector spaces, as well as the concept of linear combinations and collinearity. The lecture delves into the construction of subspaces, the generation of subspaces by polynomials, and the abstract representation of subspaces. Additionally, it discusses the concept of linear independence and provides exercises to demonstrate the properties of subspaces.