Vector Spaces: Bases and Dimensions

This lecture covers the concept of vector spaces, bases, and dimensions, including the completion of linearly independent elements into a base, the properties of subspaces, and the relationship between the dimension of a subspace and the dimension of the vector space. It also discusses the rank of a transformation, the dimension of the image and kernel of a transformation, and how to find a basis for the image. The lecture concludes with the relationship between the dimension of a vector space, the kernel, and the rank of a transformation.