Vector Spaces: Properties and Examples

This lecture covers the concept of vector spaces, starting with a focus on the properties of vector spaces and then delving into specific examples. The instructor begins by discussing the fundamental properties of vector spaces, such as addition and scalar multiplication. The lecture then progresses to examine how to determine if a given set is a vector space by verifying stability under addition and scalar multiplication. The instructor also explains the importance of understanding when two functions are equal within a vector space context. Additionally, the lecture explores the notion of subspaces within vector spaces and emphasizes the significance of verifying that a subspace is non-empty and closed under addition and scalar multiplication. The lecture concludes with a practical exercise on polynomials as algebraic objects and functions within a vector space.