example of Gram-Schmidt orthogonalization

Let us work with the standard inner product on $\mathbbmss{R}^{3}$ (dot product) so we can get a nice geometrical visualization.

Consider the three vectors

	$\displaystyle v_{1}$	$\displaystyle=(3,0,4)$
	$\displaystyle v_{2}$	$\displaystyle=(-6,-4,1)$
	$\displaystyle v_{3}$	$\displaystyle=(5,0,-3)$

which are linearly independent (the determinant of the matrix $A=(v_{1}|v_{2}|v_{3})=116\neq 0)$ but are not orthogonal.

We will now apply Gram-Schmidt to get three vectors $w_{1},w_{2},w_{3}$ which span the same subspace (in this case, all $R^{3}$ ) and orthogonal to each other.

First we take $w_{1}=v_{1}=(3,0,4)$ . Now,

$w_{2}=v_{2}-\frac{w_{1}\cdot v_{2}}{\|w_{1}\|^{2}}w_{1}$

that is,

$w_{2}=(\frac{-108}{25},-4,\frac{81}{25})$

and finally

$w_{3}=v_{3}-\frac{w_{1}\cdot v_{3}}{\|w_{1}\|^{2}}w_{1}-\frac{w_{2}\cdot v_{3}% }{\|w_{2}\|^{2}}w_{2}$

which gives

$w_{3}=(\frac{1856}{1129},\frac{3132}{1129},\frac{1392}{1129})$

and so $\{w_{1},w_{2},w_{3}\}$ is an orthogonal set of vectors such that $\langle w_{1},w_{2},w_{3}\rangle=\langle v_{1},v_{2},v_{3}\rangle$ .

If we rather consider $\{w_{1}/\|w_{1}\|,w_{2}/\|w_{2}\|,w_{3}/\|w_{3}\|\}$ then we get an orthonormal set.

Title	example of Gram-Schmidt orthogonalization
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