0 - Review of Linear Algebra (Optional)

Abstract

This chapter provides a review of linear algebra material. A matrix is a rectangular array of numbers—usually real numbers for linear programming—arranged in horizontal rows and vertical columns. The chapter describes the properties of matrix addition and scalar multiplication and the transpose of a matrix. The transpose of a matrix is obtained by merely interchanging the rows and columns of a matrix. The chapter discusses a systematic way of eliminating variables that will allow the reader to solve larger systems of equations. It is more efficient to carry out the operations on the augmented matrix of the given linear system instead of performing them on the equations of the linear system. Therefore, the augmented matrix of the given linear system is calculated and transformed to a matrix of a certain special form. The new matrix formed represents a linear system that has exactly the same solutions as the given system. The method is called Gauss–Jordan reduction. The chapter explains homogenous systems and describes trivial and nontrivial solutions.