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proof of Gram-Schmidt orthogonalization procedure
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(Proof)
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Note that, while we state the following as a theorem for the sake of logical completeness and to establish notation, our definition of Gram-Schmidt orthogonalization is wholly equivalent to that given in the defining entry.
Proof. We proceed by induction on  . In the case  , we suppose
 is a basis for the inner product space  . Letting
 , it is clear that
 , whence it follows that
 . Thus
 is an orthonormal basis for  , and the result holds for  . Now let
 , and suppose the result holds for arbitrary  . Let
 be a basis for an inner product space  . By the inductive hypothesis we may use
 to construct an orthonormal set of vectors
 such that
 . In accordance with the procedure outlined in the statement of the theorem, let
 be defined as
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First we show that the vectors
 are mutually orthogonal. Consider the inner product of
 with
 for
 . By construction, we have
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Now since
 is an orthonormal set of vectors, whence
 , each term in the summation on the right-hand side of the preceding equation will vanish except for the term where  . Thus by this and the preceding equation, we have
Thus
 is orthogonal to
 for
 , so we may take
 to have
 an orthonormal set of vectors. Finally we show that
 is a basis for  . By construction, each
 is a linear combination of the vectors
 , so we have  orthogonal, hence linearly independent vectors in the  dimensional space  , from which it follows that
 is a basis for  . Thus the result holds for  , and by the principle of induction, for all  . 
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"proof of Gram-Schmidt orthogonalization procedure" is owned by rspuzio. [ owner history (1) ]
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(view preamble)
See Also: Gram-Schmidt orthogonalization, example of Gram-Schmidt orthogonalization, inner product space, inner product, QR decomposition, normed vector space, orthogonal, orthogonal vectors, basis, span, linearly independent, orthonormal set, orthonormal basis
| Other names: |
Gram-Schmidt, orthogonalization |
| Also defines: |
Gram-Schmidt orthogonalization |
| Keywords: |
orthogonal, orthonormal, basis, inner product space, inner product |
This object's parent.
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Cross-references: linearly independent, linear combination, vanish, equation, side, summation, term, orthogonal, vectors, orthonormal set, inductive hypothesis, clear, induction, orthonormal basis, inner product, inner product space, basis, equivalent
There are 2 references to this entry.
This is version 8 of proof of Gram-Schmidt orthogonalization procedure, born on 2006-12-10, modified 2006-12-25.
Object id is 8611, canonical name is ProofOfGramSchmidtOrthogonalizationProcedure.
Accessed 4964 times total.
Classification:
| AMS MSC: | 65F25 (Numerical analysis :: Numerical linear algebra :: Orthogonalization) |
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Pending Errata and Addenda
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