OR05 Linear Systems and Linear Programs via Matrices

#linearsystems and #linearprogramming translated into matrix equations. We define basic variables, basic solutions, and show how matrices give flexibility in solving systems of linear equations and allow us to choose the basic variables. When solving Ax = b, when A is an m x n matrix of rank m, choose m linearly independent variables, and then x_B = B^{-1}b - B^{-1}Nx_N. This formalizes the dictionary and the #tableau methods (https://youtu.be/6qdSIHlABWk) using matrix notation and will be the basis of future lectures on the revised simplex algorithm. Will discuss the solutions of a non-homogeneous system Ax = b as compared to the solutions to the homogeneous system Ax = 0. Subscribe @Shahriari for more math videos at the undergraduate level. #operationsresearch 00:00 Introduction 01:27 Review: Matrix Multiplication (https://youtu.be/QlwfoStwjJU) 02:23 Review: Multiplying a matrix by a column vector, Linear Systems as Matrix Equations (https://youtu.be/HH0nOI2Lkko) 04:20 LP in standard form in matrix notation (https://youtu.be/-oJl1_WunlE) 04:55 Demand and Cost Vectors, Coefficient Matrix, Vector of decision variables 06:05 Nonstandard form LPs in matrix notation 06:47 Why can Ax = b be written as Bx_B + Nx_N = b? 08:28 If B invertible, all solutions of Ax = b are given by x_B = B^{-1}b - B^{-1}Nx_N 10:01 What is a basis matrix, basic variables, a basic solution, & a degenerate Basic Solution? 12:10 Example: Finding a basic solution 13:39 Connection: Matrix Equations vs RREF (https://youtu.be/swrVyVdS3PI) 16:23 Connection: Ax = 0 vs Ax = b, a matrix equation lens (https://youtu.be/UItv3I2lPHw) 19:46 Connection: Ax = 0 vs Ax = b, a linear transformation lens (https://youtu.be/p1G2aGia4mI) Previous OR video: https://youtu.be/-oJl1_WunlE Next OR video: https://youtu.be/JLaHEojWbGw Part of a series of videos on Introductory Operations Research. See https://www.youtube.com/playlist?list=PLpcU2wNhmPYfcT9e1d43M-DWhw2Y0vgQh Shahriar Shahriari is the William Polk Russell Professor of Mathematics at Pomona College in Claremont, CA USA Shahriari is a 2015 winner of the Mathematical Association of America's Haimo Award for Distinguished Teaching of Mathematics, and six time winner of Pomona College's Wig teaching award.