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Unformatted text preview: 3 Linear algebra and basic solutions To study linear programs, we make crucial use of some basic ideas from linear algebra. Before proceeding further, we quickly review some of these ideas. As usual, we use R n to denote the vector space of column vectors with n components. Consider an m-by- n matrix A . We say the columns of A span if the equation Ax = b has a solution for every vector b in R m . In the language of linear algebra, we would say, equivalently, that the columns of A , which we denote A 1 , A 2 , . . . , A n , “span” the vector space R m . Introducing additional columns to a matrix whose columns span does not alter the property. We say the columns of A are linearly independent if the only solution of the equation Ax = 0 is x = 0. Again using linear algebra language, this property is equivalent to the vectors A 1 , A 2 , . . . , A n being distinct and constituting a “linearly independent set” in R m . Deleting columns from a matrix whose columns are linearly independent does not alter the property. Finally, we call the matrix A invertible if its columns both span and are linearly independent. It’s easy to see that re-ordering the columns of A affects none of these properties. Linear algebra teaches us the following important facts. If the columns of A span, then n ≥ m . On the other hand, if the columns of A are linearly independent, then n ≤ m . Hence if A is invertible, it must be square. Conversely, if A is square, the following properties are equivalent: • A is invertible; • the columns of A span; • the columns of A are linearly independent; • some matrix F satisfies F A = I (the identity matrix); • a sequence of row operations reduces A to I . ( Row operations consist of multiplying a row by a nonzero constant, adding multiples of rows to other rows, and interchanging rows.) In that case, the matrix F is unique, and is called the inverse of A , written A- 1 . It also satisfies AA- 1 = I ....
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This note was uploaded on 10/31/2010 for the course ORIE 5300 at Cornell.