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Unformatted text preview: OR & IE 320 Optimization I Summer 2004 Linear Algebra Review Linear algebra is one of the most important key ingredients to linear programming. This handout aims to highlight some topics in linear algebra which will be of great use to us throughout this course. However, the handout is not comprehensive, and you are encouraged to review the material on your own. Here is a bunch of reference books:- Applied Mathematical Programming , Bradley, Hax, Magnanti, 1977, Addison-Wesley. Appendix 1 (recommended)- Introduction to Mathematical Programming , Hillier, Lieberman, 1995, McGraw-Hill, 2nd ed. Appendix 4 (not as comprehensive)- Linear Algebra and Its Applications , Strang, 1988, Harcourt Brace Jovanovich, 3rd ed. (the first two chapters) You can also find this material in any textbook about linear algebra. If you have any questions, ask me or the TAs. Vectors and Matrices Def: A matrix is a rectangular array of numbers. We will denote matrices by capital letters. If a matrix A has m rows and n columns, it is called an m n matrix, or a matrix of dimension m by n . Generally, a ij is used to denote the entry in the i th row and the j th column of the matrix A . Def: An m 1 matrix is called an m-dimensional column vector . A 1 m matrix is called an m-dimensional row vector . We will use lower-case letters to denote vectors. By convention, we will use the term vector for column vectors and row vectors will explicitly be stated. Def: The transpose of a matrix A , denoted A T , is formed by interchanging the rows and columns of A , that is, the i,j entry of A T is a ji . If A is an m n matrix, then A T is an n m matrix. Moreover, ( A T ) T = A . If u is a column vector, then u T is a row vector....
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- Fall '08