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201y05mt1

# 201y05mt1 - B U Department of Mathematics Math 201 Matrix...

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B U Department of Mathematics Math 201 Matrix Theory Summer 2005 First Midterm 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.) a) [4] Find c such that the following set of columns is a basis for R 3 : 1 1 - 1 , 2 1 0 , 1 1 c . Solution: A = 1 2 1 1 1 1 - 1 0 c - r 1 + r 2 r 2 -------→ r 1 + r 3 r 3 1 2 1 0 - 1 0 0 2 c + 1 2 r 2 + r 3 r 3 ------→ 1 2 1 0 - 1 0 0 0 c + 1 . Hence c = - 1, i.e., c R \ {- 1 } the given set of columns is a basis for R 3 . b) [4] Is the set of polynomials S = { 1 - x, 1 + x, 1 - x 2 } linearly independent? Solution: Consider a (1 - x ) + b (1 + x ) + c (1 - x 2 ) = 0 Then - cx 2 = 0 implies c = 0. So a + b = 0 and - a + b = 0 give that a = 0, b = 0. Thus S is linearly independent.

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201y05mt1 - B U Department of Mathematics Math 201 Matrix...

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