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Math Help - direct sum of subspaces

  1. #1
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    direct sum of subspaces

    Hey, I am confused about this question. I need some help getting started.

    Let V= \sum U_k where V is a vector space and U_1,...,U_j are the subspaces of V.
    Use the fact that U_1 \cap U_2 =0,..., (U_1+...+U_{j-1}) \cap U_j=0 to prove that V is the direct sum of the subspaces U_1,...,U_j.

    Where should I start with this? Thanks for any help.
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  2. #2
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    Re: direct sum of subspaces

    EDIT: sorry made a mistake in my working.

    Does anyone have any ideas on how to prove it?

    Thanks
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  3. #3
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    Re: direct sum of subspaces

    you are given a clue as to where to start (a fact to use). since i am not an unkind person i wll give you another: what is dim(U1 + U2)?

    what can you say therefore about dim((U1+U2)+U3),....,dim(U1+U2+....+Uj)?
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  4. #4
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    Re: direct sum of subspaces

    Quote Originally Posted by Deveno View Post
    you are given a clue as to where to start (a fact to use). since i am not an unkind person i wll give you another: what is dim(U1 + U2)?

    what can you say therefore about dim((U1+U2)+U3),....,dim(U1+U2+....+Uj)?
    dim(U_1+ U_2)=dim(U_1)+dim(U_2)-dim(U_1 \cap U_2)
    So
    dim(U1+U2+....+Uj)= [\sum_{k=1}^j dim(U_k)] - dim((U_1+...+U_{j-1}) \cap U_j)

    Thus dim(U_1+ U_2)=dim(U_1)+dim(U_2) and dim(U1+U2+....+Uj)= \sum_{k=1}^j dim(U_k).

    Is this correct? Where to from here?

    Thanks
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  5. #5
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    Re: direct sum of subspaces

    if dim(V) = n, what must any n-dimensional subspace be?
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  6. #6
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    Re: direct sum of subspaces

    Quote Originally Posted by Deveno View Post
    if dim(V) = n, what must any n-dimensional subspace be?
    If we let W be a subspace of V then dim(W) \le n and if dim(W)=n the W=V.

    So any n-dimensional subspace must be V.

    So in relation to my question, therefore V is the direct sum of the subspace U_1,...,U_j.

    Does this complete the proof?

    Thanks for your help Deveno!
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