Linear Transformations: Kernels and Images

This lecture covers the concepts of kernels and images of linear transformations between vector spaces, including their definitions, properties, and examples. It explains how to determine the kernel and image of a linear transformation associated with a matrix, as well as how to find the matrix representation of a linear transformation with respect to given bases. The lecture also discusses the construction of matrices associated with linear transformations, the bijectivity of certain applications, and the importance of bases in determining the matrix representation. Examples are provided to illustrate the calculations involved in determining kernels, images, and matrix representations.