Vector Spaces: Definitions and Properties

This lecture introduces the concept of vector spaces, defined as non-empty sets with addition and scalar multiplication operations. The properties of vector spaces, such as closure under addition and scalar multiplication, are discussed. The lecture covers the axioms that vector spaces must satisfy, including the existence of an identity element and inverses. Examples of vector spaces and subspaces are provided, along with demonstrations of how to verify if a set is a vector space. The lecture also explores the concept of under vector spaces and their relationship with subspaces. Various examples and applications illustrate the theoretical concepts presented.