Vector Spaces and Bases

This lecture delves into the abstract concept of vector spaces, exploring examples such as the space of polynomials of degree n or less. The notion of a subspace vector is introduced, along with the practical concept of linear independence and bases. The instructor explains the criteria for determining if a set of vectors is linearly independent and how to verify if they form a basis for a subspace. Through examples in R2 and R3, the lecture illustrates the importance of linear independence and the role of bases in generating subspaces. The discussion extends to matrices, highlighting the relationship between invertible matrices and bases in Rn. The lecture concludes with a focus on the canonical basis of Rn, emphasizing the significance of the identity matrix in defining bases.