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Unformatted text preview: B U Department of Mathematics Math 201 Matrix Theory Spring 2005 Second Midterm This archive is a property of Bogazi 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 W be the plane in R 3 defined by x- y + z = 0. a) Find an orthonormal basis for W . b) Determine the orthogonal complement W of W . Solution: a) The vectors a = (1 , 1 , 0) and b = (0 , 1 , 1) form a basis for W since they are linearly independent. To form an orthonormal basis we will use Gram-Schmidt method. As k a k = 2 take q 1 = (1 / 2 , 1 / 2 , 0). Then q 2 = b- ( q T 1 b ) q 1 = - 1 / 2 1 / 2 1 . Hence k q 2 k = 3 2 implies q 2 = (- 1 / 6 , 1 / 6 , 2 / 6). b) To find the orthogonal complement W of W it suffices to determine the Null-space of A = 1 / 2 1 / 2- 1 / 6 1...
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This note was uploaded on 04/29/2008 for the course MATH 201 taught by Professor Sadik during the Spring '08 term at Blue Mountain College.
- Spring '08