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Homework Help: Span of subspace

  1. Feb 9, 2010 #1
    1. The problem statement, all variables and given/known data

    Here's a statement, and I am supposed to show that it holds.

    If x,y, and z are vectors such that x+y+z=0, then x and y span the same subspace as y and z.


    2. Relevant equations

    N/A

    3. The attempt at a solution

    If x+y+z=0 it means that the set {x,y,z} of vectors is linearly dependent. Because of this dependence, the vectors cannot span a subspace with dimension greater than 2.
    That is, they can span subspaces with dimensions 0,1 and 2.

    • If they span a subspace with dim=0, then x=y=z=0.
    • If they span a subspace with dim=1, then two vectors are negative multiples of each other with the third one being the zero vector.
    • If they span a subspace with dim=2, then one is a linear combination (with -1 as coefficients) of the other two.

    In all these cases x and y span the same subspace as y and z.
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    Any suggestions are greatly appreciated.

    Thanks.
     
  2. jcsd
  3. Feb 10, 2010 #2

    HallsofIvy

    User Avatar
    Science Advisor

    I think you are missing the point of the question.

    Let v be in the subspace spanned by x and y. Then v= ax+ by for some numbers a and b. But x+ y+ z= 0 so x= -y- z. v= a(-y- z)+ by= (b-a)y+ (-a)z. That is, v is a linear combination of y and z and so is in the span of y and z.

    That proves that the subspace spanned by x and y is a subspace of the span of y and z. I will leave it to you to show that the span of y and z is a subspace of the span of x and y.

    Let v be in the subspace spanned by y and z, Then v= ....
     
  4. Feb 10, 2010 #3
    Ah, I certainly missed the point of the question!

    Let v be in the subspace spanned by y and z.
    Then v=ay+bz for some numbers a and b.
    But x+y+z=0 so z=-x-y.
    v=(a-b)y+(-bx), that is, v is a linear combination of y and x and so is in the span of y and x.
    This proves that the subpsace spanned by y and z is a subspace of the span of y and x.

    We have now proved that the subspace spanned by x and y is the same as the one spanned by y and z.

    Thank you!
     
    Last edited: Feb 10, 2010
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