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March 27, 2017

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Find two vectors v and w such that the three vectors u = (1,-1,-1), v and w are linearly independent independent.

  • Linear algebra - ,

    Three vectors are linearly independent if the determinant formed by the vectors (in columns) is non-zero.
    So for u=(1,-1,-1), v=(a,b,c), w=(d,e,f)
    There are many possible choices of v and w such that the determinant
    1 a d
    -1 b e
    -1 c f
    is non-zero.

    The simplest way is to create a triangular matrix such that the diagonal is all non-zero, or
    1 0 0
    -1 b 0
    -1 c f

    where b and f are non-zero, and the determinant evaluates to b*f≠0.

    Example: (b=-1,c=1,f=1)
    (1,-1,-1),(0,-1,1),(0,0,1) are linearly independent because the determinant
    1 0 0
    -1 -1 0
    -1 1 1
    evaluates to 1*(-1)*(-1)=-1 ≠ 0

    Note:
    vectors that are orthogonal to each other are linearly independent, since each cannot be a linear combination of the others.
    However, linear independent vectors need not be orthogonal. Therefore their dot products need not be zero.

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