Rank Theorem: Matrix Dimension and Subspaces

This lecture covers the rank theorem, which relates the dimensions of the kernel and image of a matrix. Starting with the characterization theorem, the instructor explains the rank of a matrix and its relationship with the dimensions of subspaces generated by the matrix's rows and columns. The lecture progresses to discuss equivalent matrices along lines, linearly independent vectors, and the conditions for invertible matrices. Demonstrations are provided to illustrate the concepts, leading to a detailed understanding of the rank theorem and its implications.