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Unformatted text preview: MATH321 HOMEWORK SOLUTIONS HOMEWORK #1 Section 1.2: Problems 1, 2, 11, 12, 13, 15, 16, 17, 18 Section 1.3: Problems 1, 8, 10, 13, 17, 19, 24, 25 Section 1.4: Problems 1, 2(a)(d), 3(a), 4(a), 6, 10 Krzysztof Galicki Problem 1.2.1 (See Answers to Selected Exercises ). Problem 1.2.2 ~ 0 = O = 0 0 0 0 0 0 0 0 0 0 0 0 . Problem 1.2.11 The axioms VS.1,2,5,6,7,8 are trivially satisfied as there is only one element in V . By definition 0 is the zero vector and it is its own inverse so that (VS.34) are likewise true. Problem 1.2.12 We know that F ( R , R ) is a vector space. The zero function f ( x ) is even. Adding two even functions gives an even function. Finally, if f ( x ) is even so is cf ( x ). Hence, VS.18 must all be satisfied (as they are already satisfied for F ( R , R )). That is, the set of even functions in F ( R , R ) forms a vector subspace. Problem 1.2.13 The axioms VS.13 are satisfied with the only possible zero vector being (0 , 1). The axiom VS.4 is not satisfied as ( a 1 , 0) has no inverse, i.e., there is no element ( b 1 ,b 2 ) such that ( b 1 ,b 2 ) + ( a 1 , 0) = (0 , 1) . The axioms VS.57 are satisfied but VS.8 fails as, for example, (1 + 1)( a 1 ,a 2 ) = 2( a 1 ,a 2 ) = (2 a 1 ,a 2 ) but ( a 1 ,a 2 ) + ( a 1 ,a 2 ) = (2 a 1 ,a 2 a 2 ) . Problem 1.2.15 No. The operation of vector addition is not affected by changing F from R to C . But scalar multiplication is not really defined on V as, for example, multiplication by i turns real vectors into vectors with imaginary complex coordinates....
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This note was uploaded on 11/30/2009 for the course MATH 115A taught by Professor Liu during the Winter '07 term at UCLA.
 Winter '07
 Liu
 Math, Linear Algebra, Algebra

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