math1030hw1sol

# math1030hw1sol - Math1030: HW1 solution HW1: Sect 1.2 p15,...

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Math1030: HW1 solution HW1: Sect 1.2 p15, Q12, Q16 Q19, Q20, Q21 12. Let E denote the set of even functions. For any f,g E , f ( - t ) = f ( t ) , g ( - t ) = g ( t ) , ( f + g )( - t ) = f ( - t ) + g ( - t ) = f ( t ) + g ( t ) = ( f + g )( t ) , t R . So ( f + g ) E . Addition is well-deﬁned in E . For any f E , c R , ( cf )( - t ) = c [ f ( - t )] = c [ f ( t )] = ( cf )( t ) . So ( cf ) E . Scalar multiplication is well-deﬁned in E . Let f,g,h E , a,b,t R , (VS 1) ( f + g )( t ) = f ( t ) + g ( t ) = g ( t ) + f ( t ) = ( g + f )( t ). (VS 2) [( f + g )+ h ]( t ) = ( f + g )( t )+ h ( t ) = f ( t )+ g ( t )+ h ( t ) = f ( t ) + ( g + h )( t ) = [ f + ( g + h )]( t ). (VS 3) Denote f 0 ( t ) = 0 , t R , then ( f + f 0 )( t ) = f ( t ) , f E . (VS 4) f E , denote ˜ f ( t ) = - f ( t ), then ˜ f ( - t ) = - f ( - t ) = - f ( t ) = ˜ f ( t ) , ˜ f E , and ˜ f + f = 0. (VS 5) (1 f )( t ) = 1 f ( t ) = f ( t ). (VS 6) [( ab ) f ]( t ) = abf ( t ) = a ( bf )( t ) = [ a ( bf )]( t ). (VS 7) [ a ( f + g )]( t ) = a ( f + g )( t ) = a [ f ( t ) + g ( t )] = af ( t ) + ag ( t ) = ( af + ag )( t ). (VS 8) [( a + b ) f ]( t ) = ( a + b ) f ( t ) = af ( t ) + bf ( t ) = ( af + bf )( t ). So E is a vector space. 16. V is a vector space over F with the usual deﬁnitions of matrix addition and scalar multiplication. Let A,B V , c F , then A + B V , cA V . So addition and scalar multiplication is well-deﬁned in V . Check ( V S 1) - ( V S 8) hold in the same way as question 12.

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## This note was uploaded on 04/30/2011 for the course MECHANICAL MSC taught by Professor Dr.rahula during the Spring '10 term at University of Moratuwa.

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math1030hw1sol - Math1030: HW1 solution HW1: Sect 1.2 p15,...

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