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Math Help - dimensions U, W, U+W

  1. #1
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    dimensions U, W, U+W

    let's say dimU = 1 and dimW = 3

    does dim(U+W) always equal dimU + dim W?

    If not, can someone please show me a counter example?

    Thankssssssss
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by jayshizwiz View Post
    let's say dimU = 1 and dimW = 3

    does dim(U+W) always equal dimU + dim W?

    If not, can someone please show me a counter example?

    Thankssssssss
    U=span{(1,1,1)}
    W=span{(2,2,2), (1,0,0),(0,1,0)}
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  3. #3
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    Thanks Also sprach Zarathustra for answering all my questions in all the different math sections.
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  4. #4
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    <br />
U=span(\vec{v}_1,\cdots,\vec{v}_p,\vec{s}_1,\cdots  ,\vec{s}_t) ~,~<br />
W=span(\vec{w}_1,\cdots,\vec{w}_q,\vec{s}_1,\cdots  ,\vec{s}_t) ~,~<br />
U\cap W = span(\vec{s}_1,\cdots,\vec{s}_t) <br />

    dim(U) = p+t, dim(W) = q+t, dim(U+W) = p+t+q, dim(U\cap W) = t,


    dim(U+W) + dim(U\cap W) = dim(U)+ dim(W)
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  5. #5
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    Quote Originally Posted by math2009 View Post
    <br />
U=span(\vec{v}_1,\cdots,\vec{v}_p,\vec{s}_1,\cdots  ,\vec{s}_t) ~,~<br />
W=span(\vec{w}_1,\cdots,\vec{w}_q,\vec{s}_1,\cdots  ,\vec{s}_t) ~,~<br />
U\cap W = span(\vec{s}_1,\cdots,\vec{s}_t) <br />

    dim(U) = p+t, dim(W) = q+t, dim(U+W) = p+t+q, dim(U\cap W) = t,


    dim(U+W) + dim(U\cap W) = dim(U)+ dim(W)
    how do we know for sure dim(U) = p+t? There may be a chance that in (\vec{v}_1,\cdots,\vec{v}_p,\vec{s}_1,\cdots,\vec{  s}_t) one of the vectors is a linear combination of the others. But I guess I understand the general thing you were saying... Thanks
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  6. #6
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    \vec{v}_i, \vec{w}_i,\vec{s}_i are linear independent.
    They form basis of U,W
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