# Orthogonalize

Orthogonalize[{v1,v2,}]

gives an orthonormal basis found by orthogonalizing the vectors vi.

Orthogonalize[{e1,e2,},f]

gives an orthonormal basis found by orthogonalizing the elements ei with respect to the inner product function f.

# Details and Options • Orthogonalize[{v1,v2,}] uses the ordinary scalar product as an inner product.
• The output from Orthogonalize always contains the same number of vectors as the input. If some of the input vectors are not linearly independent, the output will contain zero vectors.
• All nonzero vectors in the output are normalized to unit length.
• The inner product function f is applied to pairs of linear combinations of the ei.
• The ei can be any expressions for which f always yields real results.
• Orthogonalize[{v1,v2,},Dot] effectively assumes that all elements of the vi are real.
• Orthogonalize by default generates a GramSchmidt basis.
• Other bases can be obtained by giving alternative settings for the Method option. Possible settings include: "GramSchmidt", "ModifiedGramSchmidt", "Reorthogonalization", and "Householder".
• Orthogonalize[list,Tolerance->t] sets to zero elements whose relative norm falls below t.

# Examples

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## Basic Examples(1)

Find an orthonormal basis for two 3D vectors:

 In:= Out= Find the coefficients of a general vector with respect to this basis:

 In:= Out= ## Properties & Relations(6)

Introduced in 2007
(6.0)