Linear Algebra: Rank and Kernel

This lecture covers the concepts of rank and kernel in linear algebra, exploring the relationships between the rank of a matrix, the dimensions of its kernel, and the properties of linear transformations. It also delves into examples illustrating these concepts and the application of the Rank-Nullity Theorem to vector spaces. Additionally, it discusses the matrix representation of linear transformations with respect to different bases, including the change of basis formula. The lecture concludes with a study of orthogonal projections onto lines in a plane.