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Find a set of vectors that spans the subspace

  1. Sep 3, 2012 #1
    Find a set of vectors in [itex]\mathbb{R}^3[/itex] that spans the subspace
    [tex]
    S\,=\,\{\,u\,\in\,\mathbb{R}^3\,|\,u\cdot v\,=\,0\,\}
    [/tex]
    where v=<1,2,3>


    Maybe 12 hours of studying is too much and I'm fried or, maybe I'm looking for excuses. Either way...

    To solve this I'm trying to set up a matrix multiplication and augment it at zero. But, I just get a single linear equation which tells me that the only way I can have a span of this subspace is if my other vector is the zero vector <0,0,0>. I don't think that's right.

    [tex]

    \begin{bmatrix}
    a & b & c
    \end{bmatrix}

    *

    \begin{bmatrix}
    1\\2\\3
    \end{bmatrix}

    =

    \mathbf{0}


    [/tex]

    Getting [itex]a+2b+3c=0[/itex]

    Where's my issue?


    Thanks.
     
  2. jcsd
  3. Sep 4, 2012 #2
    Perhaps you should stop and think what it is you're calculating here. You are looking for a subspace which is perpendicular to a vector, so it's a plane. Maybe that will help you interpret your result.
     
  4. Sep 4, 2012 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    From a+ 2b+ 3c= 0, you have a= -2b- 3c so <a, b, c>= <-2b- 3c, b, c>.
     
  5. Sep 4, 2012 #4
    Thanks. I was looking at that and didn't take the next step of actually putting it into a vector form. Again, thanks.
     
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