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Spanning sets

  1. Jan 21, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that the following set of vectors are subspaces of R^m

    The set of all vectors (x,y,z) such that x+y+z=0 of R^3 .

    Then find a set that spans this subspace.

    2. Relevant equations

    3. The attempt at a solution

    I managed to proof that the set of vectors is a subspace by showing that it is non-empty, closed under addition and scalar multiplication. However, I have no idea how to start on part b, how do I find a spanning set for that subspace? If I am not mistaken, I have to find linear combinations.
  2. jcsd
  3. Jan 22, 2009 #2


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    Science Advisor
    Homework Helper

    Ok, name one vector in the subspace. Can you find another one that's independent of the first? Can you find a third that's not a combination of those two?
  4. Jan 22, 2009 #3
    is (1,1,1) one of the vector? I am confused.
  5. Jan 22, 2009 #4


    Staff: Mentor

    x = -y - z
    y = y
    z = z

    If you stare at this awhile, you might see two vectors staring back at you.
  6. Jan 22, 2009 #5


    Staff: Mentor

    Only if 1 + 1 + 1 = 0.
  7. Jan 22, 2009 #6
    oh, ok. Tell me if this is right. Since x = -y-z , y=y , z=z hence (-y-z , y , z) . So x(0,0,0) + y(-1,1,0) + z(-1,0,1) , So the spanning sets are (0,0,0) , (-1,1,0) , (-1,0,1) But the given answers dont include (0,0,0) . before all that, how do you know y=y and z=z ? I only know why x = -y-z .
    Last edited: Jan 22, 2009
  8. Jan 22, 2009 #7


    Staff: Mentor

    Well, your set spans the subspace, but it also does so if you remove (0, 0, 0).

    How did I know that y = y and z = z? The two equations are obviously true, aren't they?
  9. Jan 22, 2009 #8
    Ah, you got me there. They are obviously true. I was complicating stuffs, now looking back, that seemed a stupid question. OK, thanks, that did help me understand spanning sets better. Cheers
  10. Jan 22, 2009 #9


    Staff: Mentor

    If it didn't seem like a stupid question then, but now it does, I guess that means you're getting smarter, which is a good thing.
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