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Unformatted text preview: Math 340, Lecture 3 Solutions to Exam # 2 Problem 1: Give brief but precise answers to each of the following questions: (a) If S = { 1 , 2 , ..., n } , what is a permutation of S ? A permutation is a rearrangement of S , or more precisely, a mapping : S S which is onetoone. (b) If S = { 1 , 2 , ..., n } and if is a permutation of S and what is the sign of ? The sign of a permutation is (+1) if it requires an even number of interchanges needed to produce the rearrangement, and ( 1) if it requires an odd number of interchanges needed to produce the rearrangement. (c) Let A = a 1 , 1 a 1 ,n . . . . . . . . . a n, 1 a n,n be an n n matrix. What is the determinant of A ? The determinant is X sign ( ) a 1 , (1) a 2 , (2) a n, ( n ) . (d) Let A = a 1 , 1 a 1 ,n . . . . . . . . . a n, 1 a n,n be an n n matrix. What is the cofactor of the element a j,k of A ? The cofactor of a j,k is ( 1) j + k times the determinant of the ( n 1) ( n 1) matrix obtained from A by eliminating the j th row and k th columns. Problem 2: Let { v 1 ,..., v k } be vectors in a vector space V . Give brief but precise answers to each of the following questions: (a) What does it mean that a vector w V is a linear combination of the vectors { v 1 ,..., v k } ? It means that we can write w = 1 v 1 + + k v k where { 1 ,..., k } are scalars. (b) What does it mean that the vectors { v 1 ,..., v k } are linearly independent ? It means that if there are scalars { 1 ,..., k } such that 1 v 1 + + k v k = , then 1 = = k = 0 . (c) What does it mean that the vectors { v 1 ,..., v k } span the vector space V ?...
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This note was uploaded on 11/23/2008 for the course MATH 340 taught by Professor Meyer during the Spring '08 term at Wisconsin.
 Spring '08
 Meyer
 Math, Algebra, Matrices

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