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ECEN601_hw7_pfister

ECEN601_hw7_pfister - ECEN 601 Assignment 7 Problems 1 Let...

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ECEN 601: Assignment 7 Problems: 1. Let V be the subspace of R [ x ] of polynomials of degree at most 3. Equip V with the inner product ( f | g ) = integraldisplay 1 0 f ( t ) g ( t ) dt. (a) Find the orthogonal complement of the subspace of constant polynomials. (b) Apply the Gram-Schmidt process to the basis { 1 ,x,x 2 ,x 3 } . 2. Let W be a finite-dimensional subspace of an inner product space V , and let E be the orthogonal projection of V on W . Prove that ( | β ) = ( α | ) for all α , β in V . 3. Let S be a subset of an inner product space V . Show that ( S ) contains the subspace spanned by S . When V is finite-dimensional, show that ( S ) is the subspace spanned by S . 4. Let X and Y be vector spaces over the same set of scalars. Let LT [ X,Y ] denote the set of all linear transformations from X to Y . Let L and M be operators from LT [ X,Y ]. Define an addition operator between L and M as ( L + M )( x ) = L ( x ) + M ( x ) for all x X . Also define scalar multiplication by ( aL )( x ) = a ( L ( x ))
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