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linalg[GramSchmidt] - compute orthogonal vectors
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Calling Sequence
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GramSchmidt([v1, v2, ... , vn], opt)
GramSchmidt({v1, v2, ... , vn}, opt)
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Parameters
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v[1], v[2], ..., v[n]
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linearly independent vectors
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opt
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(optional) the word 'normalized'
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Description
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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.
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The vectors given must be linearly independent, otherwise the vectors returned will also be dependent.
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The vectors returned are normalized only if 'normalized' is specified.
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The command with(linalg,GramSchmidt) allows the use of the abbreviated form of this command.
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Examples
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Important: The linalg package has been deprecated. Use the superseding packages, LinearAlgebra and VectorCalculus, instead.
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 = 1, (2) = 0, (3) = 0})](/support/helpjp/helpview.aspx?si=6206/file07277/math116.png)
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![v1 := vector([1, 0, 0])](/support/helpjp/helpview.aspx?si=6206/file07277/math119.png)
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 = 1, (2) = 1, (3) = 0})](/support/helpjp/helpview.aspx?si=6206/file07277/math123.png)
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![v2 := vector([1, 1, 0])](/support/helpjp/helpview.aspx?si=6206/file07277/math126.png)
| (2) |
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 = 1, (2) = 1, (3) = 1})](/support/helpjp/helpview.aspx?si=6206/file07277/math130.png)
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![v3 := vector([1, 1, 1])](/support/helpjp/helpview.aspx?si=6206/file07277/math133.png)
| (3) |
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![{[0, 0, 1], [0, 1, 0], [1, 0, 0]}](/support/helpjp/helpview.aspx?si=6206/file07277/math140.png)
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 = 2, (2) = 2, (3) = 2})](/support/helpjp/helpview.aspx?si=6206/file07277/math144.png)
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![u1 := vector([2, 2, 2])](/support/helpjp/helpview.aspx?si=6206/file07277/math147.png)
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 = 0, (2) = 2, (3) = 2})](/support/helpjp/helpview.aspx?si=6206/file07277/math151.png)
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![u2 := vector([0, 2, 2])](/support/helpjp/helpview.aspx?si=6206/file07277/math154.png)
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 = 0, (2) = 0, (3) = 2})](/support/helpjp/helpview.aspx?si=6206/file07277/math158.png)
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![u3 := vector([0, 0, 2])](/support/helpjp/helpview.aspx?si=6206/file07277/math161.png)
| (7) |
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![GramSchmidt([u1, u2, u3])](/support/helpjp/helpview.aspx?si=6206/file07277/math165.png)
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![[[2, 2, 2], [-4/3, 2/3, 2/3], [0, -1, 1]]](/support/helpjp/helpview.aspx?si=6206/file07277/math168.png)
| (8) |
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![GramSchmidt([u1, u2, u3], normalized)](/support/helpjp/helpview.aspx?si=6206/file07277/math172.png)
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![[vector([(1/3)*3^(1/2), (1/3)*3^(1/2), (1/3)*3^(1/2)]), vector([-(1/6)*8^(1/2)*3^(1/2), (1/12)*8^(1/2)*3^(1/2), (1/12)*8^(1/2)*3^(1/2)]), vector([0, -(1/2)*2^(1/2), (1/2)*2^(1/2)])]](/support/helpjp/helpview.aspx?si=6206/file07277/math175.png)
| (9) |
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