Vector Spaces: Examples and Subspaces

This lecture introduces various examples of vector spaces, including RN with usual operations, matrices of size M-N, and polynomials. The instructor explains the properties of vector addition and scalar multiplication for polynomials, emphasizing the uniqueness of the zero vector. The lecture then delves into the concept of subspaces, focusing on the definition and properties required for a subset to be a subspace of a vector space. Through examples and exercises, the instructor demonstrates how to verify if a given set is a subspace, highlighting the importance of the zero vector and closure under addition and scalar multiplication. The lecture concludes by discussing the space spanned by a set of vectors and how it forms a subspace of the original vector space.