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Linear Algebra

  1. Aug 18, 2008 #1
    1. The problem statement, all variables and given/known data

    Let W be the subspace of R^3 spanned by the vectors
    v1=[2,1,-2] and v2=[4,0,1]
    Find a basis for the orthogonal complement of W

    2. Relevant equations


    3. The attempt at a solution

    I can do this question except for the fact when i get the matrix in form


    i get to this bit but saying the complement is equal to (col(A))=null(A^T)

    i can't perform reduced matrix stuff when there is not an odd number at the front i just can't do it can someone start of the next step for me? if so i can do the rest of the question

    it is a resit i have tomorrow and we were told today that this question is in it, so it is worth a lot for me to know it by then
  2. jcsd
  3. Aug 18, 2008 #2
    for example if someone could show me even how to reduce

    i would be fine

    i just don't understand how it is done
  4. Aug 18, 2008 #3
    ooo i might have it i just read that i can interchange rows if so i am fine as i can let it become


    which i can easily reduce :D

    grrr rows not columns
  5. Aug 18, 2008 #4


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    Homework Helper

    You don't even have to perform row reduction. It is apparent by inspection that v1 and v2 are linearly independent (one is not a multiple of the other). So what you need to do is to find an orthogonal basis for W using a well known algorithm, and use a vector operation to find the vector orthogonal to both of them.
  6. Aug 18, 2008 #5
    ok i am very confused now i have the basis


    is that right i managed to reduce it i think to
    1 0 0/0
    0 1 -4/0
  7. Aug 18, 2008 #6
    well now i am worse for wear, how would i do that? i thought the only way to find the complement to W is to use row reduction
  8. Aug 18, 2008 #7


    User Avatar
    Staff Emeritus
    Science Advisor

    The cross product of two vectors is perpendicular to both and so spans the subspace (in R3) orthogonal to the subspace spanned by the two vectors.
  9. Aug 18, 2008 #8


    User Avatar
    Homework Helper

    Oh wait, you don't even need to find an orthogonal basis for W. The cross product would do. Silly me.
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