03 - Spring 2010 CS530 Analysis of Algorithms Homework 3...

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Unformatted text preview: Spring 2010 CS530 Analysis of Algorithms Homework 3 Homework 3, due Feb 17 You must prove your answer to every question. Problems with a ( * ) in place of a score may be a little too advanced, or too challenging to most students, so I do not assign a score to them. But I will still note if you solve them. Problem 1. Which of the following is true and which one is false? Explain. (a) (2pts) Any set of vectors containing the zero vector is linearly dependent. Solution. True. Indeed, let v 1 ,..., v n be these vectors, with v 1 = 0. Then 1 v 1 + v 2 + + v n is a nonzero linear combination of the vectors giving 0. (b) (2pts) If a 1 , a 2 are linearly dependent on { b 1 , b 2 } and c is linearly dependent on { a 1 , a 2 } then c is linearly dependent on { b 1 , b 2 }. Solution. True. Indeed, if a i = j i j b j and c = i k a i then c = X j b j X k i i j . (c) (2pts) If x is linearly independent on y and y is linearly independent of z then x is linearly independent of z . Solution. False. For example, let z = x : the vector x is clearly not linearly independent of itself. (d) (2pts) If S is a linearly dependent set then each element of S is the linear combination of other elements of S . Solution. False. Consider the set { a ,0} where a 6= 0. As seen in (a) above, this set is linearly dependent. But a is not a linear combination of the vector 0 alone. (e) (2pts) If x , y and z are in the same subspace then they are linearly dependent....
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03 - Spring 2010 CS530 Analysis of Algorithms Homework 3...

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