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Unformatted text preview: Integer Programming IE418 Lecture 3 Dr. Ted Ralphs IE418 Lecture 3 1 Reading for This Lecture • N&W Sections I.4.1I.4.3 IE418 Lecture 3 2 Some Conventions If not otherwise stated, the following conventions will be followed for lecture slides during the course: • P will denote a polyhedron contained in R n . • A will denote a matrix of dimension m by n . • b will denote a vector of dimension m . • x will denote a vector of dimension n . • c will denote a vector of dimension n . • Vectors will be column vectors unless otherwise noted. • When taking the product of vectors, we will sometimes leave off the transpose. IE418 Lecture 3 3 Linear Algebra Review: Linear Independence Definition 1. A finite collection of vectors x 1 , . . . , x k ∈ R n is linearly independent if the unique solution to ∑ k i =1 λ i x i = 0 is λ i = 0 , i ∈ [1 ..k ] . Otherwise, the vectors are linearly dependent . Let A be a square matrix. Then, the following statements are equivalent: • The matrix A is invertible. • The matrix A T is invertible. • The determinant of A is nonzero. • The rows of A are linearly independent. • The columns of A are linearly independent. • For every vector b , the system Ax = b has a unique solution....
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This note was uploaded on 08/06/2008 for the course IE 418 taught by Professor Ralphs during the Spring '08 term at Lehigh University .
 Spring '08
 Ralphs

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