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Math Help - First Isomorphism Theorem

  1. #1
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    First Isomorphism Theorem

    Let S,T be subspaces of the vector spaces V ,U (respectively), and let

    T : V  U -> (V/S) (U/T) be the linear map defined by T(x, y) = (x +S, y + T). Find the kernel of T and prove that

    (V U)/(S T) (~ =) (V/S)  (W/T).

    Plx help...
    Last edited by mathbeginner; November 3rd 2010 at 07:58 PM. Reason: edit
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  2. #2
    Senior Member roninpro's Avatar
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    First, do you mean to define T(x, y) = (x +S, y + T)?

    I think that your first step is to determine the kernel of T. If T(x,y)=(0 +S, 0 +T), what can be said about x and y?
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  3. #3
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    Quote Originally Posted by roninpro View Post
    First, do you mean to define T(x, y) = (x +S, y + T)?

    I think that your first step is to determine the kernel of T. If T(x,y)=(0 +S, 0 +T), what can be said about x and y?
    is it Ker T= T(0,0)=(0+S, 0+T) =(S,T)
    therefor Ker T = (S,T)?

    but I still don't know how to prove the following....
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mathbeginner View Post
    is it Ker T= T(0,0)=(0+S, 0+T) =(S,T)
    therefor Ker T = (S,T)?

    but I still don't know how to prove the following....
    You have that x,y)\mapsto \left(x+S,y+T)" alt="F:V\boxplus U\to (V/S)\boxplus (U/T)x,y)\mapsto \left(x+S,y+T)" /> (I use \boxplus to differentiate between external and internal direct sum), this is evidently a surjective linear homomorphism. Then, we are trying to look for what values (x,y) is T(x,y)=......equals what? We're trying to look for the origin in the vector spaces V/S and U/T but these are precisely S and T respectively. So, for what x is x+S=S and for what y is y+T=T? It is fairly clear that these statements are true if and only if x\in Sand y\in T. Thus, a quick proof shows that \ker F=S\boxplus T and so by the F.I.T. \left(V\boxplus U\right)/\left(S\boxplus T\right)\cong \left(V/S\right)\boxplus\left(U/T\right)
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  5. #5
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    I don't reli get why Ker F = S X T and how to prove that???
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mathbeginner View Post
    I don't reli get why Ker F = S X T and how to prove that???
    Come on friend. Review your definitions. You're trying to show that \ker F=S\boxplus T which is equivalent to showing F(x,y)=(S,T)\Leftrightarrow (x,y)\in S\boxplus T. I give the solution below "hidden". Try your damnedest, and only look at it as a last result.
    [spoil]Clearly if (x,y)\in S\boxplus T then F(x,y)=(x+S,y+T)=(S,T) since  x+S=\left\{x+s:s\in S\right\}=S since S\leqslant V and similarly y+T=T. Conversely, suppose that $latex F(x,y)=(S,T)[/tex] then x+S=S. But, note that x\in x+S since 0\in S and so x\in S. Similarly y+T=T\implies y\in T and so F(x,y)=(S,T)\implies (x,y)\in (S,T). Combining these we get F(x,y)=(S,T)\Leftrightarrow (x,y)\in S\boxplus T[/spoil]

    EDIT: I can't figure out how to hide it...so work on the honor code haha
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  7. #7
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    Quote Originally Posted by Drexel28 View Post
    EDIT: I can't figure out how to hide it...so work on the honor code haha
    [Spoiler] [/Spoiler] You have been gone for so long that you forgot the tags!
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  8. #8
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by TheCoffeeMachine View Post
    [Spoiler] [/Spoiler] You have been gone for so long that you forgot the tags!
    Haha, I guess so!
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