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Math Help - Finding dimension of sum of subspaces

  1. #1
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    Finding dimension of sum of subspaces

    U={(x,y,z,w):x-y+2w=0}

    W={(a,b,a,0)}

    are subspaces of R^4. Determine whether the sum U+W is direct and find dim(U+W).

    I find it is not direct since, for example, (1,1,1,0) is a member of both.

    Now for the second part I use theorem Dim(U+W)=dim(U)+dim(W)-dim('union'))

    I know dim(U) and dim(W) as I earlier calculated basis for them but how do I find the dimension of the union?
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  2. #2
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    Re: Finding dimension of sum of subspaces

    Quote Originally Posted by boromir View Post
    U={(x,y,z,w):x-y+2w=0}

    W={(a,b,a,0)}

    are subspaces of R^4. Determine whether the sum U+W is direct and find dim(U+W).

    I find it is not direct since, for example, (1,1,1,0) is a member of both.

    Now for the second part I use theorem Dim(U+W)=dim(U)+dim(W)-dim('union'))

    I know dim(U) and dim(W) as I earlier calculated basis for them but how do I find the dimension of the union?
    A basis for U is

    \Mathbf{U}_1=<-2,0,0,1> and

    \Mathbf{U}_2=<1,1,0,0>

    \Mathbf{U}_3=<0,0,1,0>

    and a Basis for W is

    \Mathbf{W}_1=<1,0,1,0>

    \Mathbf{W}_2=<0,1,0,0>

    So the unions is these four basis vectors. Now just removed any dependant Vectors!
    Last edited by TheEmptySet; January 6th 2012 at 10:20 AM. Reason: missed a basis vector
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  3. #3
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    Re: Finding dimension of sum of subspaces

    So you are saying the union of the bases immediately form a spanning set and then use the theorem that says it contains a basis? Well, doing simultaneous equations I get that {U1,U2,W1,W2} is a basis, so it is in fact equal to the whole space.
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  4. #4
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    Re: Finding dimension of sum of subspaces

    Quote Originally Posted by boromir View Post
    Now for the second part I use theorem Dim(U+W)=dim(U)+dim(W)-dim('union'))
    You got that slightly wrong. It should be Dim(U+W)=dim(U)+dim(W)-dim('intersection')). Now you know that the intersection contains (1,1,1,0), so its dimension is at least 1. If the dimension of the intersection is 2 then it would have to be the whole of W (since W is fairly obviously 2-dimensional).

    From that, you should be able to pin down the dimension of U\cap W, and then the theorem will tell you the dimension of U+W.
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  5. #5
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    Re: Finding dimension of sum of subspaces

    so how is dim(U\cap W) related to dimensions of U and W? Does it have to be at most min{dim(V),dim(W)}?
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