201s04fin[1]

# 201s04fin[1] - B U Department of Mathematics Math 201...

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Unformatted text preview: B U Department of Mathematics Math 201 Matrix Theory Spring 2004 Final Exam This archive is a property of Bo˘gazi¸ ci University Mathematics Department. The purpose of this archive is to organise and centralise the distribution of the exam questions and their solutions. This archive is a non-profit service and it must remain so. Do not let anyone sell and do not buy this archive, or any portion of it. Reproduction or distribution of this archive, or any portion of it, without non-profit purpose may result in severe civil and criminal penalties. 1.) Let A = 1 1 1 2 4 . (a) Find a basis for the left nullspace of A. Solution: A T y = 1 2 1 1 4 . y 1 y 2 y 3 = ⇔ 1 2 1 2 . y 1 y 2 y 3 = y ∈ N ( A T ) iff y 1 + 2 y 3 = 0 and y 2 + 2 y 3 = 0 iff y 1 =- 2 y 3 and y 2 =- 2 y 3 = y 1 . Hence, 2 2- 1 is a basis for N ( A T ). (b) Verify that its left nullspace is orthogonal to the column space. Solution: 2 2- 1 . 1 2 = 2 + 0- 2 = 0. 2 2- 1 . 1 1 4 = 2 + 2- 4 = 0. Hence, N ( A T ) ⊥ C ( A ). 2.) Let A = 2- 1 2- 6- 2 8- 1 5 . (a) Give a LU-decomposition of A . Solution: 2- 1 2- 6- 2 8- 1 5 3 R 1 + R 2 → R 2 2- 1 2- 3 4 8- 1 5 - 4 R 1 + R 3 → R 3 2- 1 2- 3 4 3- 3 R 2 + R 3 → R...
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201s04fin[1] - B U Department of Mathematics Math 201...

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