Linear Combinations, Spanned Subspaces by a Vector Family

This lecture covers the concept of linear combinations and subspaces spanned by a family of vectors in the context of vector spaces. It explains how to define a linear combination of vectors, the form of a vector in a spanned subspace, and the properties of subspaces generated by a vector family. The lecture also introduces the notion of a subspace as the set of all linear combinations of vectors in a given set, which can be infinite. Additionally, it discusses the proposition and proof related to subspaces generated by a vector family, emphasizing the properties of these subspaces. The lecture concludes with an example illustrating the calculation of a spanned subspace by a specific vector family.