(The GramSchmidt orthogonalization procedure) IfX1,
Chapter 5, Problem 5.4.10(choose chapter or problem)
(The GramSchmidt orthogonalization procedure) IfX1, X2,...is any sequence(niteorinnite)oflinearlyindependentvectorsinanyvector space with an inner product, it can be replaced by a sequence of linear combinations that are mutually orthogonal. The idea is that at each step one subtracts off the components parallel to the previous vectors. The procedureisasfollows.First,welet Z1 = X1/X1.Second,wedene Y2 = X2 (X2, Z1)Z1 and Z2 = Y2 Y2 . Third, we dene Y3 = X3 (X3, Z2)Z2 (X3, Z1)Z1 and Z3 = Y3 Y3 , and so on. (a) ShowthatallthevectorsZ1,Z2,Z3,... areorthogonaltoeachother. (b) Apply the procedure to the pair of functions cos x + cos 2x and 3 cos x4 cos 2x in the interval (0, ) to get an orthogonal pair
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