# 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 ExampleOfGramSchmidtOrthogonalization 2013-03-22 15:03:02 2013-03-22 15:03:02 drini (3) drini (3) 5 drini (3) Example msc 65F25 ProofOfGramSchmidtOrthogonalizationProcedure