Important: The linalg package has been deprecated. Use the superseding command, LinearAlgebra[GramSchmidt], instead.

- For information on migrating linalg code to the new packages, see examples/LinearAlgebraMigration.

The function GramSchmidt computes a list or set of orthogonal vectors from a given list or set of linearly independent vectors, using the Gram-Schmidt orthogonalization process.

The vectors given must be linearly independent, otherwise the vectors returned will also be dependent.

The vectors returned are normalized only if 'normalized' is specified.

The command with(linalg,GramSchmidt) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{linalg}\right)\:$

$\mathrm{v1}≔\mathrm{vector}\left(\left[1\,0\,0\right]\right)$

$\mathrm{v1}≔\left[\begin{array}{ccc}1& 0& 0\end{array}\right]$

$\mathrm{v2}≔\mathrm{vector}\left(\left[1\,1\,0\right]\right)$

$\mathrm{v2}≔\left[\begin{array}{ccc}1& 1& 0\end{array}\right]$

$\mathrm{v3}≔\mathrm{vector}\left(\left[1\,1\,1\right]\right)$

$\mathrm{v3}≔\left[\begin{array}{ccc}1& 1& 1\end{array}\right]$

$\mathrm{GramSchmidt}\left(\left\{\mathrm{v1}\,\mathrm{v2}\,\mathrm{v3}\right\}\right)$

$\left\{\left[0\,0\,1\right]\,\left[0\,1\,0\right]\,\left[1\,0\,0\right]\right\}$

$\mathrm{u1}≔\mathrm{vector}\left(\left[2\,2\,2\right]\right)$

$\mathrm{u1}≔\left[\begin{array}{ccc}2& 2& 2\end{array}\right]$

$\mathrm{u2}≔\mathrm{vector}\left(\left[0\,2\,2\right]\right)$

$\mathrm{u2}≔\left[\begin{array}{ccc}0& 2& 2\end{array}\right]$

$\mathrm{u3}≔\mathrm{vector}\left(\left[0\,0\,2\right]\right)$

$\mathrm{u3}≔\left[\begin{array}{ccc}0& 0& 2\end{array}\right]$

$\mathrm{GramSchmidt}\left(\left[\mathrm{u1}\,\mathrm{u2}\,\mathrm{u3}\right]\right)$

$\left[\left[2\,2\,2\right]\,\left[-\frac{4}{3}\,\frac{2}{3}\,\frac{2}{3}\right]\,\left[0\,\mathrm{-1}\,1\right]\right]$

$\mathrm{GramSchmidt}\left(\left[\mathrm{u1}\,\mathrm{u2}\,\mathrm{u3}\right]\,\mathrm{normalized}\right)$

$\left[\left[\begin{array}{ccc}\frac{\sqrt{3}}{3}& \frac{\sqrt{3}}{3}& \frac{\sqrt{3}}{3}\end{array}\right]\,\left[\begin{array}{ccc}-\frac{\sqrt{8}\sqrt{3}}{6}& \frac{\sqrt{8}\sqrt{3}}{12}& \frac{\sqrt{8}\sqrt{3}}{12}\end{array}\right]\,\left[\begin{array}{ccc}0& -\frac{\sqrt{2}}{2}& \frac{\sqrt{2}}{2}\end{array}\right]\right]$