# Wolfram Language & System 10.0 (2014)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# Orthogonalize

Orthogonalize[{v1,v2,}]
gives an orthonormal basis found by orthogonalizing the vectors .

Orthogonalize[{e1,e2,},f]
gives a basis for the orthonormal with respect to the inner product function f.

## Details and OptionsDetails 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 .
• The can be any expressions for which f always yields real results.
• Orthogonalize[{v1,v2,},Dot] effectively assumes that all elements of the 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: , , , and .
• Orthogonalize[list,Tolerance->t] sets to zero elements whose relative norm falls below t.

## ExamplesExamplesopen allclose all

### Basic Examples  (1)Basic Examples  (1)

Find an orthonormal basis for two 3D vectors:

 Out[1]=

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

 Out[2]=