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Student[LinearAlgebra]

 GramSchmidt
 compute an orthonormal set of Vectors

 Calling Sequence GramSchmidt(V, options)

Parameters

 V - list or set of Vector(s) options - (optional) parameters; for a complete list, see LinearAlgebra[GramSchmidt]

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{LinearAlgebra}]\right):$
 > $\mathrm{w1}≔⟨2,1,0,-1⟩:$
 > $\mathrm{w2}≔⟨1,0,2,-1⟩:$
 > $\mathrm{w3}≔⟨0,-2,1,0⟩:$
 > $B≔\mathrm{GramSchmidt}\left(\left[\mathrm{w1},\mathrm{w2},\mathrm{w3}\right]\right)$
 ${B}{≔}\left[\left[\begin{array}{c}\frac{{1}}{{3}}{}\sqrt{{6}}\\ \frac{{1}}{{6}}{}\sqrt{{6}}\\ {0}\\ {-}\frac{{1}}{{6}}{}\sqrt{{6}}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {-}\frac{{1}}{{6}}{}\sqrt{{2}}\\ \frac{{2}}{{3}}{}\sqrt{{2}}\\ {-}\frac{{1}}{{6}}{}\sqrt{{2}}\end{array}\right]{,}\left[\begin{array}{c}\frac{{2}}{{21}}{}\sqrt{{21}}\\ {-}\frac{{4}}{{21}}{}\sqrt{{21}}\\ {-}\frac{{1}}{{21}}{}\sqrt{{21}}\\ {0}\end{array}\right]\right]$ (1)
 > ${B}_{1}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{B}_{1},{B}_{1}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{B}_{2},{B}_{1}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{B}_{3}$
 ${1}{,}{0}{,}{0}$ (2)
 > $\mathrm{GramSchmidt}\left(\left[⟨1|0⟩,⟨1|b⟩\right]\right)$
 $\left[\left[\begin{array}{cc}{1}& {0}\end{array}\right]{,}\left[\begin{array}{cc}{0}& \frac{{b}}{\sqrt{{{b}}^{{2}}}}\end{array}\right]\right]$ (3)