Linear Equations: Vectors and Matrices

This lecture delves into the fundamental concepts of linear equations, vectors, and matrices. The instructor starts by discussing systems of linear equations and their solutions, emphasizing the use of vectors and operations like addition and scalar multiplication. The lecture progresses to cover the composition of operations on vectors, leading to the concept of linear combinations. Through examples and graphical interpretations, the lecture explores how different combinations of vectors generate new vectors in a space. The instructor then introduces the matrix-vector product and demonstrates the distributive property of matrix multiplication. The lecture concludes with a detailed explanation of solving systems of linear equations using matrices, including homogenous systems and the concept of parametric solutions.