IOE510+PS+03 - affine subspace .] a) Consider the following...

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IOE510 Winter 2010 Problem Set 3 Due Wednesday, January 27, 2010 by 9am All problems should be uploaded electronically to CTools. Problem 1 [All vectors are written here as row vectors, rather than column vectors, simply for convenience. Feel free to use column vectors in your write-up!] a) Does the set of vectors [1 0 1], [1, 2, 3], and [1 -1 -1] span R 3 ? Is it a basis for R 3 ? Be sure to justify your answers. b) Does the set of vectors [1 0 1], [1 2 3], [ 3 2 1], and [1 -1 1] span R 3 ? Is it a basis for R 3 ? Be sure to justify your answers. Problem 2 Problem 3 Problem 4 Bertsimas and Tsitsiklis problem 1.20 (a)
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Problem 5 [The purpose of this problem is to better understand the meaning of an
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Unformatted text preview: affine subspace .] a) Consider the following polyhedron P: {x R 2 | 3 x 1 + 2 x 2 = 4} Prove that P is an affine subspace by identifying a subspace S and a vector b such that for any vector x in S , the vector (x + b ) is in P. b) Is your choice of b unique? If yes, prove it. If no, define the set of all valid vectors b . c) Consider the following polyhedron P: {x R n | Ax = b} where A is an m-x-n matrix and b is a column vector in R m . Prove that P is an affine subspace by identifying a subspace S and the set of vectors {b }such that for any vector x in S , the vector (x + b ) is in P. d) For what matrices A will there only be one valid vector b ?...
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This note was uploaded on 04/11/2010 for the course IOE 510 taught by Professor Staff during the Fall '08 term at University of Michigan.

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IOE510+PS+03 - affine subspace .] a) Consider the following...

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