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Unformatted text preview: B U Department of Mathematics Math 201 Matrix Theory Spring 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 nonprofit 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 nonprofit purpose may result in severe civil and criminal penalties. 1.) a) Find c such that the following set of columns is a basis for R 3 : 1 1 1 , 2 1 , 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 1 2 c + 1 2 r 2 + r 3 → r 3→ 1 2 1 1 c + 1 . Hence c 6 = 1, i.e., ∀ c ∈ R \ { 1 } the given set of columns is a basis for R 3 . b) 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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This note was uploaded on 11/16/2011 for the course MATH 201 taught by Professor Soysal during the Fall '08 term at Boğaziçi University.
 Fall '08
 SOYSAL
 Math

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