innerProductSpaces1

# innerProductSpaces1 - 2 Calculating THEOREM projwu and u...

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2 2. Calculating proj w u and u – w u. THEOREM Proof Let w 1 = k w, where k is a scalar 12 2 2 22 1 () ( ) , since is perpendiculat to , and 0. So, k= ,and proj kk k k == = + = = === w u w w + ww w + w w w w uw w w ii i i i i i i i i i If u and w are vectors in 2-spase or 3-space and if w 0, then 2 proj = (vector component of along ) proj = (vector component of orthogonal to ) = −− w w w u w w uu u w u w i i i
3 3 . Orthonormal Bases The solution of a problem is often greatly simplified by choosing a basis in which the vectors are orthogonal to one another. In this section we shall show how such bases can be obtained . DEFINITION EXAMPLE 1 Constructing an Orthonormal Set Let u 1 = (0,1,0), u 2 = (1,0, 1), u 3 =(1,0,-1) and assume that R 3 has the Euclidean inner product. The set of vectors S = { u 1 , u 2 , u 3 ) is orthogonal since < u 1 , u 2 > = 0 < u 1 3 < u 2 3 A set of vectors in an inner product space is called an orthogonal set if all pairs of distinct vectors in the set are orthogonal. An orthogonal set in which each vector has norm 1 is called orthonormal.

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4 The Euclidean norms of the vectors are:

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## This note was uploaded on 06/20/2008 for the course AM 025b taught by Professor Kiruscheva during the Winter '07 term at UWO.

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innerProductSpaces1 - 2 Calculating THEOREM projwu and u...

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