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Math Help - Orthogonal Subspaces

  1. #1
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    Orthogonal Subspaces

    Let S be the subspace of R4 containing all vectors with x1 + x2 + x2 + x4 = 0. Find a basis for the space S orthogonal, containing all vectors orthogonal to S.
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  2. #2
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    There are a few ways to solve this but first note that basis for S is

    (x_1,x_2,x_3,-x_1-x_2-x_3)=x_1(1,0,0,-1)+x_2(0,1,0,-1)+x_3(0,0,1,-1)

    So you could just find a vector perpendicular to these three above vectors.

    Or if you take the gradient of x_1+x_2+x_3+x_4=0 you get

    (1,1,1,1) you can check using the vectors above that this is orthogonal to the space S.
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    I don't know about the gradient, but could you tell me how to find a vector that is perpendicular to those three?
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  4. #4
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    You have three vectors a, b and c.
    Vector p is perpendicular to a, b and c if
    ap=0
    bp=0
    cp=0
    You have 3 equations and 4 unknown in p.
    So one coordinate in p you may choose.
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  5. #5
    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by veronicak5678 View Post
    I don't know about the gradient, but could you tell me how to find a vector that is perpendicular to those three?
    Look at x1 + x2 + x2 + x4 = 0 as a equation of plane in R^4, gradient is the perpendicular vector to that plane ( the coefficients of x1,x2,x3,x4, which is the vector: (1,1,1,1))
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