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Find vectors that span a subspace

  1. Dec 14, 2011 #1
    1. The problem statement, all variables and given/known data
    Find a pair of vectors that span the subspace x+y-2z=0 of R3.


    2. Relevant equations
    x+y-2z=0


    3. The attempt at a solution
    I just guessed some numbers since its such a simple equation and came up with (1,-1,0) and (2,0,1). I was just wondering what the standard method is to figure this out.. Thanks!
     
  2. jcsd
  3. Dec 14, 2011 #2

    radou

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    Homework Helper

    Well, if a vector is in your subspace, the it must be of the form (2z-y, y, z). If you "extract" y and z as parameters, you have (2z-y, y, z) = y(-1, 1, 0) + z(2, 0, 1). Since (2z-y, y, z) is arbitrarily chosen, it follows that the set {(-1, 1, 0), (2, 0, 1)} spans your subspace, since we have a "rule" on how to obtain the coefficients of the linear combination of these two vectors. Even more, this set is linearly independent, and is actually a basis for your subspace, which is even stronger that what you needed to find.
     
  4. Dec 14, 2011 #3
    AH! thank you so much! that was driving me crazy!!
     
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