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The GramSchmidt(V) command computes a list or set of orthonormal Vectors by using the Gram-Schmidt orthogonalization process. If V is an empty list or set, GramSchmidt(V) returns an empty list or set, respectively.

The number of Vectors returned is the dimension of the vector space spanned by V. In particular, if the Vectors in V are not linearly independent, fewer Vectors than the number in V are returned.

The dimension and orientation of all Vectors in V must be the same.

By default in the Student[LinearAlgebra] package, complex conjugates are not used when forming dot products, including when applying the Gram-Schmidt process. This behavior can be modified with the SetDefault command.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right)\:$

$\mathrm{w1}≔⟨2\,1\,0\,-1⟩\:$

$\mathrm{w2}≔⟨1\,0\,2\,-1⟩\:$

$\mathrm{w3}≔⟨0\,-2\,1\,0⟩\:$

$B≔\mathrm{GramSchmidt}\left(\left[\mathrm{w1}\,\mathrm{w2}\,\mathrm{w3}\right]\right)$

$B≔\left[\left[\begin{array}{c}\frac{\sqrt{6}}{3}\\ \frac{\sqrt{6}}{6}\\ 0\\ -\frac{\sqrt{6}}{6}\end{array}\right]\,\left[\begin{array}{c}0\\ -\frac{\sqrt{2}}{6}\\ \frac{2\sqrt{2}}{3}\\ -\frac{\sqrt{2}}{6}\end{array}\right]\,\left[\begin{array}{c}\frac{2\sqrt{21}}{21}\\ -\frac{4\sqrt{21}}{21}\\ -\frac{\sqrt{21}}{21}\\ 0\end{array}\right]\right]$

$B\left[1\right]·B\left[1\right],B\left[1\right]·B\left[2\right],B\left[1\right]·B\left[3\right]$

$1,0,0$

$\mathrm{GramSchmidt}\left(\left[⟨1|0⟩\,⟨1|b⟩\right]\right)$

$\left[\left[\begin{array}{cc}1& 0\end{array}\right]\,\left[\begin{array}{cc}0& \frac{b}{\sqrt{b^2}}\end{array}\right]\right]$