hw1-solutions

hw1-solutions - HOMEWORK ASSIGNMENT 1: SOLUTIONS TO THE...

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Unformatted text preview: HOMEWORK ASSIGNMENT 1: SOLUTIONS TO THE GRADED EXERCISES MATH 115A, SECTION 3, FALL 2011 Ex. 2 (*) Let S be any nonempty set and F be any field, and let F ( S , F ) denote the set of all functions from S to F . Two functions f and g in F ( S , F ) are called equal if f ( s ) = g ( s ) for each s P S . For f , g P F ( S , F ) and c P F , let f + g , cf P F ( S , F ) be the functions defined by ( f + g )( s ) = f ( s ) + g ( s ) ( s P S ) , ( cf )( s ) = cf ( s ) ( s P S ) . Prove the following statements. (a) The operations + and . defined above give F ( S , F ) the structure of a vector space over F . Solution. (VS1) For f , g P F ( S , F ) , f + g = g + f because for each s P S , ( f + g )( s ) = f ( s ) + g ( s ) by definition, and f ( s ) + g ( s ) = g ( s ) + f ( s ) because F is a field; finally g ( s ) + f ( s ) = ( g + f )( s ) by definition. Since this holds for every s P S , we have f + g = g + f . (VS2) is proved in a similar way. The zero vector of (VS3) is the function 0 P F ( S , F ) defined by 0 ( s ) = 0 for every s P S , and we then have f + = f for every f because f ( s ) + ( s ) = f ( s ) + = f ( s ) for every s P S , since F is a field. For (VS4), given f , we let f be the function given by ( f )( s ) = ( f ( s )) for s P S , and then we will have f + ( f ) = because f ( s ) + ( f ( s )) = 0 for every s P S . For (VS5), ( 1 f )( s ) = 1 f ( s ) = f ( s ) for every s P S , and then 1...
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hw1-solutions - HOMEWORK ASSIGNMENT 1: SOLUTIONS TO THE...

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