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InnerProd-Feb15-11LARGE

# InnerProd-Feb15-11LARGE - Math 425 Spring 2011 Jerry L...

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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 h , i X and h , i Y . Example: X = R 4 and Y = R 7 . Vectors x , z X are orthogonal if h x , z i X = 0. Let L : X Y be a linear map. Then the adjoint map L * : Y X is deﬁned by the property h Lx , y i Y = h x , L * y i 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 h , i X etc, we’ll write h , i 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

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InnerProd-Feb15-11LARGE - Math 425 Spring 2011 Jerry L...

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