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Finding the Orthogonal Complement

  1. Apr 15, 2016 #1
    1. The problem statement, all variables and given/known data

    Let V be the 2-dim subspace of R^3 spanned by V1 = (1,1,1) and V2 = (-2,0,1). Find the orthogonal compliment Vperpendicular.
    2. Relevant equation

    X + Y + Z = 0
    -5X + Y + 4Z = 0


    3. The attempt at a solution
    Firstly, I orthogonalize the basis for V and get the vectors (1,1,1) and (-5,1,4).

    Then I apply the dot product and end up with the above equations. I found that the solution space for this system of equations is a single vector (x,y,z) = (.5z, -1.5z,z) . Usually I will get 2 vectors and then orthogonalize them as the basis for the orthogonal compliment. Can this single vector (.5,-1.5,1) be the orhogonal compliment? Or did I make a mistake somewhere? Please help.
     
  2. jcsd
  3. Apr 15, 2016 #2

    Samy_A

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    (x,y,z) = (.5z, -1.5z,z) doesn't represent a single vector, but the one dimensional subspace spanned by the vector (.5,-1.5,1).
     
    Last edited: Apr 15, 2016
  4. Apr 15, 2016 #3

    blue_leaf77

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    You don't need that. You just need to solve the system of equations
    $$
    x+y+z=0 \\
    -2x+z=0
    $$
    Obviously, there will be at least one free variable, make it ##z## and solve the other variables in terms of ##z##.
     
  5. Apr 15, 2016 #4

    ehild

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    In R3, the orthogonal complement of a two-dimensional subspace is one-dimensional, that is spanned by a single vector. You have found that vector, and the subspace consists of all vectors of form (x,y,z) = (.5z, -1.5z,z) , as you wrote. So your solution is correct, but it was not needed to orthogonalize the the basis. You can find a vector perpendicular to other two independent ones by cross-product them.
     
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