Dimension Theory: Vector Spaces and Bases

This lecture covers the dimension theory of vector spaces, introducing concepts such as cent-free families, bases, generating families, and linear applications. The instructor explains the conditions for a family to be a basis, the extraction and completion theorems, and the existence of bases under the axiom of choice. The lecture also delves into linear applications, ranks, and the properties of injective, surjective, and bijective linear transformations.