Linear Combinations and Span

This lecture covers the concepts of linear combinations and span in the context of matrix equations. The instructor explains how to determine if a matrix equation has at least one solution by analyzing the conditions on the matrix A and the vector B. Through examples, the lecture demonstrates how to express a vector as a linear combination of other vectors and how to use the span to describe the set of solutions. The lecture also explores the implications of having pivots in each row of matrix A and the equivalence between different properties of a matrix. Additionally, the lecture delves into the concept of homogeneous systems and how to generalize solutions using the span, illustrating the geometric interpretation of solution sets.