2.6 Dual Spaces 2 - 2.6. Trapezoidal rule - R b a p ( t )...

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Unformatted text preview: 2.6. Trapezoidal rule - R b a p ( t ) dt ( b- a ) f ( b )+ f ( a ) 2 Simpsons rule - R b a f ( t ) dt ( b- a ) 6 ( f ( a ) + 4 f ( a + b 2 ) + f ( b )) 11. For x V, x 7 2 T ( x ) = [ T ( x ) x 7 T tt 1 ( x ) = T tt ( b x ) = b xT t To show commuting, [ T ( x ) = b xT t in W ** : W * F g W * , i.e. g : W F a linear functional Show ( b xT t )( g ) = [ T ( x )( g ) [ T ( x )( g ) = g ( T ( x )) = ( gT )( x ) = widehatx ( gT ) = b x ( T t g ) = b xT t ( g ) 2 T = T tt 1 12. ( ) = ** = { x 1 ,x 2 , ,x n } a basis for V * = { x * 1 ,x * 2 , ,x * n } a basis for V * , where x * i : V F a linear functional s.t. f i ( x j ) = ij Since V ** is the dual space of V * ** = { x ** 1 ,x ** 2 , ,x ** n } a basis for V ** s.t. x ** i : V * F * linear functional s.t. x ** i ( x * j ) = ij Show x ** i = b x i , i, ( ) = { x 1 ,x 2 , ,x n } b x ( x * i ) = ij = x ** i ( x * j ) , i,j 130 PNU-MATH 2.6....
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This note was uploaded on 02/14/2012 for the course MAS 4105 taught by Professor Rudyak during the Spring '09 term at University of Florida.

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2.6 Dual Spaces 2 - 2.6. Trapezoidal rule - R b a p ( t )...

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