Linear Independence: Definition and Examples

This lecture covers the definition of linear independence in vector spaces, focusing on the unique solution of vector equations to determine if a set of vectors is linearly independent or dependent. Examples are provided to illustrate the concept, including scenarios where matrices associated with the vectors have no free variables. The lecture also discusses the implications of linear dependence between columns of a matrix, emphasizing the relationship with non-trivial solutions of homogeneous equations. Special cases and theorems related to linear independence are presented, highlighting conditions for vectors to be linearly dependent. Additionally, the concept of linear transformations as linear applications between vector spaces is introduced.