lecture 4 - Math 482 (Lecture 4): The simplex method I:...

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Math 482 (Lecture 4): The simplex method I: linear algebra reminders and basic (feasible) solutions Today: We're going to start discussing the elements needed in the simplex method. Reminders about linear algebra : Linear algebra is concerned about solving systems of linear equations Ax =b where A is an mxn matrix in n unknowns. * The basic tool is Gaussian elimination as encoded by augmented matrices . * The rank of the matrix A is the number of linearly independent columns it has. This is at most m. Assumption: A actually has rank m. (Otherwise there are some redundant equations we can remove/or inconsistent equations.) * Now I'm going to start introducing terminology that's specific to describing the simplex method. Definition: A basis of A is a collection of m linearly independent columns 1≤ j 1 < j 2 ≤... ≤ j m of our matrix A. In class exercise: Consider the matrix 1 1 1 0 1 0 What are the bases? What are the nonbases? If an mxn matrix has at least one basis, can you always solve Ax=b? For example, can you solve 1 1 0 1 0 1 1 0 1 Brief solution: Columns {1,4,5}, {1,2,4}, {1,2,5}, {2,3,4}, {2,3,5}, {2,4,5} are bases. Non bases are all other choices of columns, e.g., {1,2,3}. You can determine this by either row reduction or taking a determinant. When you have a basis, you can always solve Ax=b. One way to say this is that the span of the basis is all of R m .
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Another way to think of this is to construct a particular basic solution : set all non basis column variables to zero and use row reduction. The next example discusses the notions of basic solution and basic feasible solutions (see
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This note was uploaded on 02/19/2012 for the course MATH 482 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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lecture 4 - Math 482 (Lecture 4): The simplex method I:...

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