A3S - ASSIGNMENT 3 Solutions For i = 1 ,...,k , let A i be...

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Unformatted text preview: ASSIGNMENT 3 Solutions For i = 1 ,...,k , let A i be an n i × n i +1 matrix whose rows span R n i +1 . 1. [3 marks] Prove that the rows of A 1 ··· A k span R n k +1 . We have ( A 1 ··· A k ) T = A T k ··· A T 1 . It suffices to show that the columns of the above matrix span R n k +1 . This is equivalent to showing that A T k ··· A T 1 x = b has a solution for any b ∈ R n k +1 . Let b ∈ R n k +1 be given. We apply the following chain of reasoning: Since the columns of A T k span R n k +1 , A T k x = b has a solution y k ∈ R n k . Since the columns of A T k- 1 span R n k , A T k- 1 x = y k has a solution y k- 1 ∈ R n k- 1 . Since the columns of A T k- 2 span R n k- 1 , A T k- 2 x = y k- 1 has a solution y k- 2 ∈ R n k- 2 . . . . Since the columns of A T 2 span R n 3 , A T 2 x = y 3 has a solution y 2 ∈ R n 2 . Since the columns of A T 1 span R n 2 , A T 1 x = y 2 has a solution y 1 ∈ R n 1 ....
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This note was uploaded on 04/07/2010 for the course MATH 136 taught by Professor All during the Fall '08 term at Waterloo.

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A3S - ASSIGNMENT 3 Solutions For i = 1 ,...,k , let A i be...

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