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Sum of two subspaces - question.

  1. Dec 2, 2012 #1
    1. The problem statement, all variables and given/known data
    Is it possible to add the following subspaces: W_1 = Sp{(1,0,0)} and W_3 = Sp{(0,1,-1), (0,0,1)}?


    2. Relevant equations



    3. The attempt at a solution
    Will their sum be: Sp{(1,1,-1),(1,0,1)}?
     
  2. jcsd
  3. Dec 2, 2012 #2

    Mark44

    Staff: Mentor

    No.

    It would help to think about the geometry here. What does the W1 subspace look like? Can you describe it geometrically?

    What does the W3 subspace look like? Can you describe it geometrically?

    Are any vectors in W1 also in W3? Are any vectors in W3 also in W1?

    The "sum" of the two sets (actually the direct sum, W1[itex] \oplus[/itex] W3) is the set of vectors v such that v = w1 + w3, where w1 ##\in## W1 and w3 ##\in## W3.
     
  4. Dec 2, 2012 #3
    I understand that their sum is a direct sum as the intersection is null, but I am not sure how that helps me to find Sp(W_1+W_3).
     
  5. Dec 2, 2012 #4
    I was also asked to prove that (W_1 intersection W_2) + (W_1 intersection W_3) = W_1 intersection (W_2+W_3), where W_1, W_2, W_3 are subspaces in vector space V.

    Attempt at solution:
    Let W_1=Sp(K), W_2=Sp(U), W_3=Sp(L)
    Hence, (W_1 intersection W_2) = Sp(K) intersection Sp(U) = Sp(K intersection U)
    Hence, (W_1 intersection W_3) = Sp(K) intersection Sp(L) = Sp(K intersection L)
    Hence, (W_1 intersection W_2) + (W_1 intersection W_3) = Sp(K intersection U) + Sp(K intersection L) = Sp(K intersection U + K intersection L) = Sp(K intersection (U+L))
    I am really not sure this is correct. Is it? Is this really how I should try proving it?
     
  6. Dec 2, 2012 #5
    Mark44, their geometrical representation will be:
    W_3 = y-(-z) plane and z axis, W_1 = x axis
     
  7. Dec 2, 2012 #6
    So W1 contains all vectors of the form (x,0,0), and W3 contains all vectors of the form (0,y,z).

    What vectors can be expressed as a sum of one vector from W1 and one from W3?
     
  8. Dec 2, 2012 #7
    R^3.
    How may I check whether W_1+W_2=V, where V is a vector space and W_1 and W_2 are two subspaces in V, and W_1=(t,s,t-2s,2t) where t,s belong to R, and W_2=(x,y,z,-x-y-z)?
    I first tried to write W_1 as {(1,0,1,2), (0,1,-2,0)} and W_2 as {(1,0,0,-1), (0,1,0,-1),(0,0,1,-1)} and then concluded that since dim(W_1+W_2) was not equal to dim(V) they could not possibly be equal. But I am really not sure that's correct. Could anyone please advise?
     
  9. Dec 2, 2012 #8

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    There's a systematic way to do these problems. Set up a matrix where the rows are your vectors and row reduce it. When you are done the remaining nonzero vectors will be a basis for W_1+W_2.
     
    Last edited: Dec 2, 2012
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