PS #6 SOLUTION - .secirtam n n erauqs fo nnM tes eht no...

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Math 3013 Problem Set 6 Problems from § 3.1 (pgs. 189-190 of text): 11,16,18 Problems from § 3.2 (pgs. 140-141 of text): 4,8,12,23,25,26 1. (Problems 3.1.11 and 3.1. 16 in text). Determine whether the given set is closed under the usual operations of addition and scalar multiplication, and is a (real) vector space. (a) The set of all diagonal n × n matrices. Let A =[ a ij ] be a diagonal n × n matrix and λ a real number. Then λ A = λ a 11 0 ··· 0 0 a 22 0 . . . . . . . . . 00 a nn = λa 11 0 0 0 λa 0 . . . . . . . . . λa nn is also diagonal. So the set of diagonal n × n matrices is closed under scalar multiplication. Let A a ij ] and B b ij ] be two diagonal n × n matrices. Then A + B = a 11 0 0 0 a 22 0 . . . . . . . . . a nn + b 11 0 0 0 b 0 . . . . . . . . . b = a 11 + b 0 0 0 a + b 22 0 . . . . . . . . . a + b nn is also diagonal. So the set of diagonal n × n matrices is also closed under vector addition. (b) The set P n of all polynomials in x , with real coefficients and of degree less than or equal to n , together with the zero polynomial. Let p = a n x n + a n - 1 x n - 1 + + a 1 x + a 0 be a polynomial of degree n and let λ be a real number. Then λp = λa n x n + λa n - 1 x n - 1 + + λa 1 x + λa 0 is also a polynomial of degree n . Hence, the set P n is closed under scalar multiplication. Let p = a n x n + a n - 1 x n - 1 + + a 1 x + a 0 p == a n x n + a n - 1 x n - 1 + + a 1 x + a 0 be two polynomials in P n . Then p + p =( a n + a n ) x n + ( a n - 1 + a n - 1 ) x n - 1 + +( a 1 + a 1 ) x a 0 + a 0 ) is also a polynomial of degree n . So the set P n is closed under vector addition. 2. (Problem 3.1.18 in text). Determine whether the following statements are true or false. (a) Matrix multiplication is a vector space operation on the set M m × n of m × n matrices. False. Vector space operations are just scalar multiplication and vector addition. (b) Matrix multiplication is a vector space operation on the set M n × n of square n × n matrices.
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This note was uploaded on 07/04/2011 for the course MATH 3013 taught by Professor Staff during the Spring '08 term at Oklahoma State.

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PS #6 SOLUTION - .secirtam n n erauqs fo nnM tes eht no...

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