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Math Help - Graham-Schmidt Orthogonalisation

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    Graham-Schmidt Orthogonalisation

    Write down a basis for the solution space of the equation x - y + 2z - w = 0.
    Use the Gram-Schmidt orthogonalisation procedure to nd an orthonormal basis of
    the solution space. (Use the standard Euclidean inner product, or dot product.)

    Firstly, to write down the a basis for the solution space, can I just choose any four vectors that satisfy the given equation?

    Secondly, when I do choose any vectors ie (1,1,1,2) and (5,8,3,3) I always get really messy fractions doing the orthogonalisation. Is there any way to know which vectors to choose to make it neat?

    Cheers,
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    Quote Originally Posted by U-God View Post
    Write down a basis for the solution space of the equation x - y + 2z - w = 0.
    Use the Gram-Schmidt orthogonalisation procedure to find an orthonormal basis of
    the solution space. (Use the standard Euclidean inner product, or dot product.)

    Firstly, to write down the a basis for the solution space, can I just choose any four vectors that satisfy the given equation?

    Secondly, when I do choose any vectors ie (1,1,1,2) and (5,8,3,3) I always get really messy fractions doing the orthogonalisation. Is there any way to know which vectors to choose to make it neat?
    Firstly, the solution space is three-dimensional, so you will only need three (linearly independent) vectors, not four, for a basis.

    Secondly, if you choose simple vectors (preferable with some coordinates zero) you are likely to minimise the arithmetic complexity. For example, you could choose (1,1,0,0), (0,0,1,2) and (1,0,0,1) as the basis.

    But the GS procedure is inherently messy, and even if you start with simple-looking vectors you are still liable to end up with some complicated fractions.
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