Linear Transformations: Matrices and Applications

This lecture covers linear transformations, matrices, and their applications. It explains how to determine the matrix associated with a linear transformation, the concepts of surjective, injective, and bijective transformations, and the properties of symmetric transformations. The lecture also discusses the equivalence between linear transformations and matrices, as well as the implications of injectivity and surjectivity in the context of matrices and solutions. Additionally, it explores the relationships between columns of matrices, linear independence, and reduced row-echelon form. Various theorems are presented to illustrate the interplay between linear transformations and matrices.