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InnerProd-Feb15-11-2up

InnerProd-Feb15-11-2up - Math 425 Spring 2011 Jerry L...

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Math 425, Spring 2011 Jerry L. Kazdan Inner Product Summary This is a summary of some items from class on Tues, Feb. 15, 2011. S ETTING : Linear spaces X , Y with inner products ( , ) X and ( , ) Y . Example: X = R 4 and Y = R 7 . Vectors x , z X are orthogonal if ( x , z ) X = 0. Let L : X Y be a linear map. Then the adjoint map L : Y X is defined by the property ( Lx , y ) Y = ( x , L y ) X for all x X , y Y . Observation: ( LM ) = M L . For real matrices, the adjoint is just the transpose. For complex matrices, it is complex conjugate transpose. Instead of writing ( , ) X etc, we’ll write ( , ) since the inner product being used will be obvious. In L 2 ( a , b ) on functions f with f ( a ) = 0 and f ( b ) = 0, if L : = d dx , then L = d dx . If one ignores the boundary conditions (that is, forget the boundary terms when integrating by parts), one gets the formal adjoint . P ROJECTION AND O RTHOGONAL D ECOMPOSITION . Let V X be a linear subspace. If x X , write x = v + z , where v V , z V .
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