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Math Help - Problem help - Orthogonal Bases

  1. #1
    Jun 2010

    Problem help - Orthogonal Bases

    Consider the following Subspace on R3:

    U={(x1,x2,x3) R3: x1-2x2+x3=0}

    Which of the following group of vectors is an orthogonal basis of U?

    A - {(1,-2,1), (-1,0,1)}
    B - {(-1,0,1), (1,1,1)}
    C - {(2,1,0), (-1,0,1)}
    D - {(2,1,0), (0,0,1)}
    E - {(1,-2,1), (-1,0,1), (1,1,1)}
    Can anyone give me a detailed resolution for this exercise?
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  2. #2
    Junior Member slider142's Avatar
    May 2009
    Brooklyn, NY
    First note that the equation describes a plane in R^3. Specifically, recall that a plane may be specified by the set of all points whose displacement vectors from a proprietary point are normal to a specific vector, called the normal vector. Normality is defined by the dot product vanishing, so note that your equation may be rewritten as (1, -2, 1)\cdot [(x_1, x_2, x_3) - (0, 0, 0)] = 0 A normal vector to your plane is thus (1, -2, 1).
    The two basis vectors must have a dot product of 0 to be orthogonal and must have a cross product that is a scalar multiple of the normal vector, since they must lie in that plane.
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