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Vector spaces

  1. Nov 20, 2007 #1
    Suppose V and W are vector spaces, and {v1....vn} is basis for V and T. S is an element of L(V, W). Suppose further that T(vi)=S(vi) for all i with 1<= i <=n. Show that S=T.

    Here's what I think.

    Because S is an element of L(V,W), S:V-->W means that S has a basis of {v1...vn}, and two vector spaces that form a bijective linear map (which S and T do because they have the same basis) are isomorphic. Moreover, because T(vi)=S(vi) then by their isomorphism, T and S must be equal.

    This is the last question on my practice midterm and I'm unsure if this is how to proceed with the proof. Any comments, especially on how I could do this more "formally"?
     
  2. jcsd
  3. Nov 20, 2007 #2
    1. The problem statement, all variables and given/known data

    Suppose V and W are vector spaces, and {v1....vn} is basis for V and T. S is an element of L(V, W). Suppose further that T(vi)=S(vi) for all i with 1<= i <=n. Show that S=T.



    2. Relevant equations

    none


    3. The attempt at a solution

    Here's what I think.

    Because S is an element of L(V,W), S:V-->W means that S has a basis of {v1...vn}, and two vector spaces that form a bijective linear map (which S and T do because they have the same basis) are isomorphic. Moreover, because T(vi)=S(vi) then by their isomorphism, T and S must be equal.

    This is the last question on my practice midterm and I'm unsure if this is how to proceed with the proof. Any comments, especially on how I could do this more "formally"?
     
    Last edited: Nov 20, 2007
  4. Nov 21, 2007 #3

    Dick

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    Maps (in L(V,W)) don't have bases. Vector spaces have bases. Can you rephrase the question knowing that?
     
    Last edited: Nov 21, 2007
  5. Nov 21, 2007 #4

    morphism

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    (I suppose you also meant T is in L(V,W) as well.)

    What do you mean when you say "S has a basis"?! S is a linear map, not a vector space. V is what has the basis {v_1, ..., v_n}. Now to show that S=T, it will be enough to show that T(v)=S(v) for all v in V. But each v in V has a unique representation in terms of the basis vectors. So use this (and don't forget that T and S are linear).
     
  6. Nov 21, 2007 #5

    HallsofIvy

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    ?? You mean, I think, that "{v1,..., vn} is a basis for V." and that "T and S are elements of L(V,W).

    As Dick pointed out, linear maps do not have a basis! You mean that {v1,...,vn} is a basis for V (which you had already said). In order to prove that T= S, you must prove that T(v)= S(v) for any vector v in V. v can be written as a linear combination of the basis vectors: v= a1v1+ ...+ a2v2. What happens if you apply both T and S to that?
     
    Last edited: Nov 21, 2007
  7. Nov 21, 2007 #6

    HallsofIvy

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    This was also posted in the "homework" area so I am merging the two threads.

    mrroboto, do NOT double post!
     
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