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Student[LinearAlgebra][GramSchmidt] - compute an orthonormal set of Vectors
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Calling Sequence
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GramSchmidt(V, options)
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Description
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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.
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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.
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The dimension and orientation of all Vectors in V must be the same.
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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.
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Examples
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![with(Student[LinearAlgebra])](/support/helpjp/helpview.aspx?si=4392/file04994/math86.png)
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![B := GramSchmidt([w1, w2, w3])](/support/helpjp/helpview.aspx?si=4392/file04994/math102.png)
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*6^(1/2)], [(1/6)*6^(1/2)], [0], [-(1/6)*6^(1/2)]]), Vector[column]([[0], [-(1/6)*2^(1/2)], [(2/3)*2^(1/2)], [-(1/6)*2^(1/2)]]), Vector[column]([[(2/21)*21^(1/2)], [-(4/21)*21^(1/2)], [-(1/21)*21^(1/2)], [0]])]](/support/helpjp/helpview.aspx?si=4392/file04994/math105.png)
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![B[1].B[1], B[1].B[2], B[1].B[3]](/support/helpjp/helpview.aspx?si=4392/file04994/math109.png)
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![GramSchmidt([`<|>`(1, 0), `<|>`(1, b)])](/support/helpjp/helpview.aspx?si=4392/file04994/math116.png)
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, Vector[row]([0, b/(b^2)^(1/2)])]](/support/helpjp/helpview.aspx?si=4392/file04994/math119.png)
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