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Unformatted text preview: Math 115a Final exam Lecture 1 Fall 2009 Name: Instructions: There are 10 problems. Make sure you are not missing any pages. Unless stated otherwise, you may use without proof anything proven in the sections of the book covered by this test. You may only cite an exercise from the book if it was assigned as homework. You must prove your answers (OF COURSE!). Question Points Score 1 10 2 15 3 10 4 15 5 15 6 10 7 10 8 15 9 10 10 15 Total: 125 1. (10 points) Let A be an upper triangular matrix. Use the definition of the determinant to prove that det( A ) is the product of the diagonal entries of A . Solution We proceed by induction. The claim is easy to show for small matrices. Suppose it is true for upper triangular ( n 1) ( n 1) matrices. Let A = a 11 a 1 n . . . . . . a nn . Recall that if A is n n , then det A = n X i =1 ( 1) 1+ i a i 1 det( f A i 1 ) , where f A i 1 is the ( i, 1)cofactor of A . Since A is upper triangular, a i 1 = 0 if i 2. Hence det A = n X i =1 ( 1) 1+ i a i 1 det( f A i 1 ) = a 11 det( g A 11 ) . But g A 11 is an upper triangular ( n 1) ( n 1) matrix, so the induction hypothesis applies, giving us det( g A 11 ) = a 22 a 33 a nn . Hence det A = a 11 a 22 a 33 a nn . 2. (15 points) Let V be a finite dimensional inner product space. Let W be a subspace of V , and let { w 1 ,...,w k } be an orthonormal basis of W . Define T ( x ) = k X j =1 h x,w j i w j . (a) (5 points) Prove N ( T ) = W (b) (5 points) Prove R ( T ) = W . (c) (5 points) Compute T * ( x ) for any x in V . Solution (a) Suppose T ( x ) = 0. Then k j =1 h x,w j i w j = 0. Since { w 1 ,...,w k } is orthonormal, it is linearly independent, which implies that h x,w j i = 0 for j = 1 ,...k . This proves N ( T ) W . On the other hand, if x W , then h x,w j i = 0 for j = 1 ,...k , which implies that T ( x ) = k j =1 h x,w j i w j = 0. This proves W N ( T )....
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This note was uploaded on 02/22/2010 for the course MATH Math 115 taught by Professor Taft during the Fall '06 term at UCLA.
 Fall '06
 Taft
 Math, Linear Algebra, Algebra

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