InnerProd-Feb15-11

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

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Unformatted text preview: Math 425, Spring 2011 Jerry L. Kazdan Inner Product Summary This is a summary of some items from class on Tues, Feb. 15, 2011. SETTING: Linear spaces X , Y with inner products h , i X and h , i X . 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 = h x , L * y i 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. 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 . PROJECTION AND ORTHOGONAL DECOMPOSITION. Let V ⊂ X be a linear subspace. If x ∈ X , write x = v + z , where v ∈ V , z ⊥ V ....
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InnerProd-Feb15-11 - Math 425 Spring 2011 Jerry L Kazdan...

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