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Math Help - Collection of all linear transformation is a vector space

  1. #1
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    Collection of all linear transformation is a vector space

    Let V and W be vector space over a field F, and let T,U: V -> W be linear. Prove that the collection of all linear transformation from V to W is a vector space over F.

    Proof.

    Let F denote the set of all linear transformations from V to W.
    Let T,U,S be in F, and let t,u,s be in V.

    Commutativity of addition: (T+U)(t+u)=T(t)+U(u)=U(u)+T(t)=(U+T)(u+t)

    Is this the right way to do it? Thanks
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Let V and W be vector space over a field F, and let T,U: V -> W be linear. Prove that the collection of all linear transformation from V to W is a vector space over F.

    Proof.

    Let F denote the set of all linear transformations from V to W.
    Let T,U,S be in F, and let t,u,s be in V.

    Commutativity of addition: (T+U)(t+u)=T(t)+U(u)=U(u)+T(t)=(U+T)(u+t)

    Is this the right way to do it? Thanks
    There is a lot of stuff to show here. Let T_1,T_2 be linear transformations V\mapsto W. Define T_1 + T_2 to be the sum (as a function sum) of the linear transformations. Then (T_1 + T_2 )(\bold{u}+\bold{v}) = T_1 (\bold{u}) + T_1(\bold{v}) + T_2(\bold{u})+T(\bold{v}) = (T_1+T_2)(\bold{u}) + (T_1+T_2)(\bold{v}). Futhermore, (T_1+T_2)(k\bold{u}) = T_1(k\bold{u}) + T_2(k\bold{u}) = kT_1(\bold{u})+kT_2(\bold{u}) = k(T_1+T_2)(\bold{u}). Thus, the set of all linear transformations is closed under this definition. There are more things we need to prove. We need to show that this set S forms an abelian group under this operation +. We know that T_1 + T_2 = T_2 + T_1 and (T_1+T_2)+T_3 = T_1 + (T_2 + T_3) and T_1 + \hat{0} = \hat{0}+T_1 = T_1 (where \hat{0} is the trivial linear transformation). And if -T_1 is the negative linear transformation then T_1 + (-T_1) = \hat{0}. Which shows S is an abelian group. To show that S is a vector space we need to confirm more things. Can you finish that?
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  3. #3
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    Yes, sorry I was lazy to type the other stuff out, I know that I have to prove all those, but I wasn't sure if I define the linear transformations right.

    Thanks!
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