computational and applied math

# computational and applied math - CAAM 336 DIFFERENTIAL...

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CAAM 336 · DIFFERENTIAL EQUATIONS 1. [25 points: 5 points per part] For all problems below, assume V is a vector space endowed with an inner product, ( · , · ). (a) Suppose W = span { φ 1 } is a one-dimensional subspace of V . Write down a formula for the best approximation to v V from the subspace W . In many circumstances, we are given a linearly independent basis φ 1 ,...,φ N for the subspace W = span { φ 1 ,...,φ N } , and would like to create an orthogonal basis ψ 1 ,...,ψ N for this same subspace W . This procedure is called the Gram–Schmidt process: Set ψ 1 = φ 1 . For k = 2 ,...,N Let b φ k denote the best approximation to φ k from span { ψ 1 ,...,ψ k - 1 } . Set ψ k = φ k - b φ k . end (b) Use the Gram–Schmidt process to construct orthogonal basis vectors ψ 1 2 3 for the subspace W = span { φ 1 2 3 } of R 4 , where φ 1 = 1 0 0 0 , φ
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## This note was uploaded on 01/21/2012 for the course CAAM 330 taught by Professor Tompson during the Fall '09 term at UVA.

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computational and applied math - CAAM 336 DIFFERENTIAL...

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