Vector Spaces: Bases and Dimension

This lecture covers the concept of bases and dimension in vector spaces. It explains the theorem of extracted base, stating that if a vector set is linearly dependent, then it generates a subspace. The lecture also discusses the uniqueness of writing vectors as linear combinations, the consequences of having a known dimension, and the proposition related to the dimension of subspaces. Additionally, it introduces the concept of the rank of a matrix and its relation to linear independence. Examples and proofs are provided to illustrate the theoretical concepts.