Vector Spaces: Definitions and Applications

This lecture covers the fundamental concepts of vector spaces, including internal and external composition laws, scalar multiplication, and the verification of vector space properties. It also explores examples of vector spaces, such as sets of vectors in 2D and 3D, functions defined on real numbers, and polynomials with real coefficients. The lecture delves into subspaces, linear applications, and linear maps, emphasizing the preservation of properties under addition and scalar multiplication. Additionally, it discusses the evaluation map and provides exercises to reinforce the understanding of vector space concepts.