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Math Help - Find Basis orthogonal to a given u

  1. #1
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    Find Basis orthogonal to a given u

    Let u = (1 , 0, 2, 1).
    Find a basis for the subspace of vectors orthogonal to u in R4.

    Please help me get started. The ones I did before used the Gram- Schmidt process, however I only have one vector, so I am unsure of how to begin.

    Thanks.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Those vectors (x_1, x_2, x_3, x_4) which are orthogonal to (1,0,2,1) satisfy the equation x_1+2x_3+x_4=0. Now you can let, say, x_1=u, x_2=v, x_3=w and then we have x_4=-u-2w. Thus your subspace consists of all vectors of the form (u, v, w, -u-2w). It's a 3-dimensional vector space (3 parameters) so you'll need 3 vectors in your basis; by picking values of u,v,w it's easy to find 3 vectors who do the job. Take it from there!
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  3. #3
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    Ok, I think I got it now.
    Example, if I were to let u = 1 and v = w= 0. then x4 would be -1 and that would be one of the vectors in the basis.
    So a solution would be

    ( 1 0 0 -1)
    (0 1 0 0)
    (0 0 1 -2), am I right? I am unsure if the second one is correct, but I think it makes sense to me.
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  4. #4
    MHF Contributor Bruno J.'s Avatar
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    That's good!
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  5. #5
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    Thank you soo much
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