s2 - C O350 L INEAR P ROGRAMMING - S OLUTIONS TO ASSIGNMENT...

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CO350 LINEAR PROGRAMMING - SOLUTIONS TO ASSIGNMENT 2 Exercise 1. Let { a 1 ,...,a k } ⊂ R m be a set of linearly independent vectors. (a) Prove that if a k +1 R m is not in the subspace spanned by a 1 ,...,a k then the set { a 1 ,...,a k +1 } is linearly independent. (b) Prove that the set { a 1 ,...,a k } is contained in a basis of R m . Solution: (a) Suppose by contradiction that there exist coefficients c 1 ,...,c k +1 R , not all zero, such that c 1 a 1 + c 2 a 2 + ... + c k +1 a k +1 = 0 . As a 1 ,...,a k are linearly independent, c k +1 6 = 0 . Then a k +1 = - 1 c k +1 ( c 1 a 1 + c 2 a 2 + ... + c k a k ) . Therefore a k +1 is in the subspace spanned by a 1 ,...,a k , contradiction. (b) The statement is trivial if { a 1 ,...,a k } is a basis of R m . If not, then there exists a vector a k +1 R m which is not in the subspace spanned by a 1 ,...,a k . By part (a), the set { a 1 ,...,a k +1 } is linearly independent. We can iterate this process until we have m linearly independent vectors (hence a
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This note was uploaded on 05/28/2011 for the course CO 350 taught by Professor S.furino,b.guenin during the Fall '07 term at Waterloo.

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s2 - C O350 L INEAR P ROGRAMMING - S OLUTIONS TO ASSIGNMENT...

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