Projections - Projections With the inner product we...

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Projections With the inner product <x,y> , we have angles ( <x,y> = k x k 2 k y k 2 cos ( θ )), and can speak of orthogonality: x y ⇐⇒ <x,y> = 0. Here we will consider the standard inner product for R n : <x,y> x t y , but more general inner products can be very useful in many applications and algorithm development. If S is a subspace of R n (write S R n ), we say x S if x is orthogonal to every element of S . The subspace S R n is called the orthogonal complement of S S ≡ { x R n : x S } , and R n = S S is a direct sum decomposition of R n into complementary subspaces in such a way that each x R n has the unique factorization x = u + v , with u S and v S . In this setting, u is the orthogonal projection of x onto S . In this note, we will be looking at a transformation P , which satisfies x R n , Px = u, the orthogonal projection of x onto S. If
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This note was uploaded on 12/18/2010 for the course PHYS 5073 taught by Professor Mark during the Fall '10 term at Arkansas.

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