example of Gram-Schmidt orthogonalization

Let us work with the standard inner product on 3 (dot productMathworldPlanetmath) so we can get a nice geometrical visualization.

Consider the three vectors

v1 =(3,0,4)
v2 =(-6,-4,1)
v3 =(5,0,-3)

which are linearly independentMathworldPlanetmath (the determinantMathworldPlanetmath of the matrix A=(v1|v2|v3)=1160) but are not orthogonalMathworldPlanetmath.

We will now apply Gram-Schmidt to get three vectors w1,w2,w3 which span the same subspacePlanetmathPlanetmathPlanetmath (in this case, all R3) and orthogonal to each other.

First we take w1=v1=(3,0,4). Now,


that is,


and finally


which gives


and so {w1,w2,w3} is an orthogonal set of vectors such that w1,w2,w3=v1,v2,v3.

If we rather consider {w1/w1,w2/w2,w3/w3} then we get an orthonormal setMathworldPlanetmath.

Title example of Gram-Schmidt orthogonalizationPlanetmathPlanetmath
Canonical name ExampleOfGramSchmidtOrthogonalization
Date of creation 2013-03-22 15:03:02
Last modified on 2013-03-22 15:03:02
Owner drini (3)
Last modified by drini (3)
Numerical id 5
Author drini (3)
Entry type Example
Classification msc 65F25
Related topic ProofOfGramSchmidtOrthogonalizationProcedure