PlanetMath: example of Gram-Schmidt orthogonalization

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example of Gram-Schmidt orthogonalization

(Example)

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

which are linearly independent (the determinant of the matrix $A=(v_1\vert v_2\vert v_3)=116\neq 0)$ but are not orthogonal.

$\includegraphics{vectores1.eps}$

We will now apply Gram-Schmidt to get three vectors

which span the same subspace (in this case, all

) and orthogonal to each other.

First we take . Now,

$\displaystyle w_2= v_2 - \frac{w_1\cdot v_2}{\Vert w_1\Vert^2}w_1$

that is,

$\displaystyle w_2 = (\frac{-108}{25},-4,\frac{81}{25})$

and finally

$\displaystyle w_3=v_3-\frac{w_1\cdot v_3}{\Vert w_1\Vert^2}w_1 - \frac{w_2\cdot v_3}{\Vert w_2\Vert^2}w_2$

which gives

$\displaystyle w_3=(\frac{1856}{1129},\frac{3132}{1129},\frac{1392}{1129})$

and so $\{w_1,w_2,w_3\}$ is an orthogonal set of vectors such that $\langle w_1,w_2,w_3\rangle=\langle v_1,v_2,v_3\rangle$ .

$\includegraphics{vectores2}$

If we rather consider $\{w_1/\Vert w_1\Vert,w_2/\Vert w_2\Vert,w_3/\Vert w_3\Vert\}$ then we get an orthonormal set.

"example of Gram-Schmidt orthogonalization" is owned by drini. [ owner history (1) ]

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65F25 (Numerical analysis :: Numerical linear algebra :: Orthogonalization)

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