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2.6 Dual Spaces 2 - 2.6 Trapezoidal rule R b a p t dt ≈ b...

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Unformatted text preview: 2.6. Trapezoidal rule - R b a p ( t ) dt ≈ ( b- a ) f ( b )+ f ( a ) 2 Simpson’s 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→ ψ 2 T ( x ) = [ T ( x ) x 7→ T tt ψ 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 ) ∴ ψ 2 T = T tt ψ 1 12. ψ ( β ) = β ** β = { x 1 ,x 2 , ··· ,x n } a basis for V ⇒ β * = { x * 1 ,x * 2 , ··· ,x * n } a basis for V * , where x * i : V → F a linear functional s.t. f i ( x j ) = δ ij Since V ** is the dual space of V * ∃ β ** = { x ** 1 ,x ** 2 , ··· ,x ** n } a basis for V ** s.t. x ** i : V * → F * linear functional s.t. x ** i ( x * j ) = δ ij Show x ** i = b x i , ∀ i, ψ ( β ) = { x 1 ,x 2 , ··· ,x n } b x ( x * i ) = δ ij = x ** i ( x * j ) , ∀ i,j 130 PNU-MATH 2.6....
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2.6 Dual Spaces 2 - 2.6 Trapezoidal rule R b a p t dt ≈ b...

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