Gram-Schmidt Orthogonalization Process

Gram-Schmidt Orthogonalization ProcessP. Sam JohnsonNovember 16, 2014P. Sam Johnson (NITK) Gram-Schmidt Orthogonalization ProcessNovember 16, 20141 / 31 DefinitionLet V be an inner product space, x,y V . Let A,B be subsets of V . x,y = 0(we write x y)x and y areorthogonalto each otherx y for every pair of distinct vectorsx,y in AA isorthogonalA is orthogonal and every vector in A hasnorm1A isorthonormalevery vector in A is orthogonal to everyvector in BA isorthogonalto BDefinitionLet S be a subspace of an inner product space. We say that B is anorthogonal basis(resp. anorthonormal basis) of S if B is a basis of Sand B is an orthonormal (resp.)

2 that is orthogonal to v 1, put it in the basis. If V contains a nonzero vector v 3 that is orthogonal to v 1 and v 2, put it in the basis. Proceed in this way. The chosen points v 1;v 2;::: will be mutually orthogonal. The generated set is an orthogonal set, which is also a linearly independent. Thus, if V is n dimensional, the selection ...

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  Process, Gram, Schmidt, Orthogonal, Gram schmidt orthogonalization process, Orthogonalization

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