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Thread: Tricky question, subspaces dimensions

  1. #1
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    Tricky question, subspaces dimensions

    i need to disprove the following with an example:

    Let $\displaystyle U_1,U_2,U_3$ be subspaces of $\displaystyle V$.

    $\displaystyle dim(U_1+U_2+U_3)=dim(U_1)+dim(U_2)+dim(U_3)-dim(U_1\cap U_2)-dim(U_1\cap U_3)-dim(U_2\cap U_3)+dim(U_1\cap U_2\cap U_3)$

    i used Venn Diagram and covered any possible variations of $\displaystyle U_1,U_2$ and $\displaystyle U_3$ but nothing seems to work.

    Thanks in advanced!
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  2. #2
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    Re: Tricky question, subspaces dimensions

    suppose V = R3, and set:

    U1 = span({(1,1,0)})

    U2 = span({(1,0,0)})

    U3 = span({(0,1,0)}).

    then dim(U1+U2+U3) = 2

    but dim(U1) + dim(U2) + dim(U3) - dim(U1∩U2) - dim(U1∩U3) - dim(U2∩U3) + dim(U1∩U2∩U3)

    = 1 + 1 + 1 - 0 - 0 - 0 + 0 = 3
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  3. #3
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    Re: Tricky question, subspaces dimensions

    Presumably, this question arose from the true statement that $\displaystyle dim(U_1+U_2)=dim(U_1)+dim(U_2)-dim(U_1\cap U_2)$. When you try to extend this to 3 subspaces, you need to find $\displaystyle (U_1+U_2)\cap U_3$, which is definitely not equal to $\displaystyle (U_1\cap U_3)+(U_2\cap U_3)$. Basically, this is why Deveno's counter example above works. Notice though if $\displaystyle U_1\subset U_3$ or $\displaystyle U_2\subset U_3$, then $\displaystyle (U_1+U_2)\cap U_3=(U_1\cap U_3)+(U_2\cap U_3)$, and so in this case your original statement is true. (It boils down to the true statement about two subspaces.)
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    Re: Tricky question, subspaces dimensions

    thanks man!

    when i used Venn Diagram i always considered $\displaystyle dim(U_1+U_2+U_3)$ as being equal to $\displaystyle dim(V)$.
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    Re: Tricky question, subspaces dimensions

    Quote Originally Posted by johng View Post
    Presumably, this question arose from the true statement that $\displaystyle dim(U_1+U_2)=dim(U_1)+dim(U_2)-dim(U_1\cap U_2)$. When you try to extend this to 3 subspaces, you need to find $\displaystyle (U_1+U_2)\cap U_3$, which is definitely not equal to $\displaystyle (U_1\cap U_3)+(U_2\cap U_3)$. Basically, this is why Deveno's counter example above works. Notice though if $\displaystyle U_1\subset U_3$ or $\displaystyle U_2\subset U_3$, then $\displaystyle (U_1+U_2)\cap U_3=(U_1\cap U_3)+(U_2\cap U_3)$, and so in this case your original statement is true. (It boils down to the true statement about two subspaces.)
    exactly so: we might have $\displaystyle u_1 \in U_1, u_2 \in U_2$ and $\displaystyle u_1+u_2 \in U_3$, but that does not guarantee that $\displaystyle u_1,u_2 \in U_3$.
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