04-14-03 - Lecture 27 Andrei Antonenko April 14, 2003 1...

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Unformatted text preview: Lecture 27 Andrei Antonenko April 14, 2003 1 Orthogonal bases In this section we will generalize the example from the previous lecture. Let { v 1 ,v 2 ,...,v n } be an orthogonal basis of the Euclidean space V . Our goal is to find coordinates of the vector u in this basis, i.e such numbers a 1 ,a 2 ,...,a n , that u = a 1 v 1 + a 2 v 2 + + a n v n . The familiar way is to write a linear system, and solve it. But since the vectors of the basis are orthogonal, we can do the following. First, lets multiply the expression above by v 1 . Well get: h u,v 1 i = a 1 h v 1 ,v 1 i + a 2 h v 1 ,v 2 i + + h v 1 ,v n i . But all products h v 1 ,v 2 i ,..., h v 1 ,v n i are equal to 0, so well have h u,v 1 i = a 1 h v 1 ,v 1 i , and thus a 1 = h u,v 1 i h v 1 ,v 1 i . In the same way multiplying by v 2 ,v 3 ,...,v n we will get formulae for other coefficients: a 2 = h u,v 2 i h v 2 ,v 2 i , ..., a n = h u,v n i h v n ,v n i . Definition 1.1. The coefficients defined as a 1 = h u,v 1 i h v 1 ,v 1 i , ..., a n = h u,v n i h v n ,v n i . are called Fourier coefficients of the vector u with respect to basis { v 1 ,v 2 ,...,v n } . Moreover, we proved the following theorem: Theorem 1.2. Let { v 1 ,v 2 ,...,v n } be an orthogonal basis of the Euclidean space V . Then for any vector u , u = h u,v 1 i h v 1 ,v 1 i v 1 + h u,v 2 i h v 2 ,v 2 i v 2 + + h u,v n i h v n ,v n i v n This expression is called Fourier decomposition and can be obtained in any Euclidean space, e.g. the space of continuous functions C [ a,b ]. 1 2 Projections In this lecture we will continue study orthogonality. Well start now with the projection of a vector to another vector. ' ' ' ' ' ' ' ' * A A K ' ' ' ' ' * w v u cw = proj w v The projection of the vector v along the vector w is the vector proj w v = cw proportional to w , such that u = v- cw is orthogonal to...
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04-14-03 - Lecture 27 Andrei Antonenko April 14, 2003 1...

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