Linear Equations: Matrix Notation & Solutions

This lecture introduces matrix-vector product notation for expressing linear combinations efficiently. It covers rewriting systems of linear equations in matrix form, the augmented matrix, and the concept of spanning columns. The instructor demonstrates how to solve homogeneous systems of equations, showing that they always have at least one solution. The lecture also explores the geometric interpretation of solutions, such as lines and planes in higher dimensions. Additionally, it delves into the properties of matrices, including distributivity, and the equivalence of different statements regarding matrices. The concept of free variables and the parametric representation of solutions are illustrated through examples, highlighting the connection between matrices, vectors, and solutions.