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Math Help - Help with linear transformation / subspace proof

  1. #1
    Senior Member Danneedshelp's Avatar
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    Help with linear transformation / subspace proof

    Q: Let T:V\rightarrow{W} be a linear transformation, and let U be a subspace of W. Prove that the set T^{-1}(U)=\{\vec{v}\in{V} | T(v)\in{U}\} is a subspace of V. What is T^{-1}(U) if U=\{\vec{0}\} ?

    A: I dont have much, so I will show my train of thought...

    First, I want to show T^{-1}(U) is nonempty. I can do this by simply noting U is a subspace; therefore, T^{-1}(U) is nonempty. I don't know if I am reading the set correctly. Do I wan't to show the range of T^{-1}(U) is nonempty?

    I am confused by what T^{-1}(U) represents.

    Does it mean:

    T(\vec{v})=c_{1}T(\vec{v_{1}})+c_{2}T(\vec{v_{2}})  +<br />
...+c_{k}T(\vec{v_{k}})\in{U}

    T^{-1}(T(\vec{v}))=\vec{v}=c_{1}\vec{v_{1}}+c_{2}\vec{  v_{2}}+<br />
...+c_{k}\vec{v_{k}}\in{V}

    ???

    Thanks
    Last edited by Danneedshelp; July 27th 2009 at 11:04 PM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Danneedshelp View Post
    Q: Let T:V\rightarrow{W} be a linear transformation, and let U be a subspace of W. Prove that the set T^{-1}(U)=\{\vec{v}\in{V} | T(v)\in{V}\} is a subspace of V. What is T^{-1}(U) if U=\{\vec{0}\} ?

    A: I dont have much, so I will show my train of thought...

    First, I want to show T^{-1}(U) is nonempty. I can do this by simply noting U is a subspace; therefore, T^{-1}(U) is nonempty. I don't know if I am reading the set correctly. Do I wan't to show the range of T^{-1}(U) is nonempty?

    I am confused by what T^{-1}(U) represents.

    Does it mean:

    T(\vec{v})=c_{1}T(\vec{v_{1}})+c_{2}T(\vec{v_{2}})  +<br />
...+c_{k}T(\vec{v_{k}})\in{U}

    T^{-1}(T(\vec{v}))=\vec{v}=c_{1}\vec{v_{1}}+c_{2}\vec{  v_{2}}+<br />
...+c_{k}\vec{v_{k}}\in{V}

    ???

    Thanks

    T^{-1}U is the inverse image of U under T. That is it is the set of all elements x of W such that Tx\in U.

    That T^{-1}U is non-empty is shown as T0_V=0_W that is for any linear transformation the zero vector maps to the zero vector. Then since 0_W\in U we have 0_V\in T^{-1}U

    To show that T^{-1}U is a subspace of V it is sufficient to show that for any scalars \alpha_1, \alpha_2 and x_1, x_2 \in T^{-1}U that:

     <br />
\alpha_1x_1+\alpha_2x_2 \in T^{-1}U<br />

    CB
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  3. #3
    Senior Member Danneedshelp's Avatar
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    Quote Originally Posted by CaptainBlack View Post

    To show that T^{-1}U is a subspace of V it is sufficient to show that for any scalars \alpha_1, \alpha_2 and x_1, x_2 \in T^{-1}U that:

     <br />
\alpha_1x_1+\alpha_2x_2 \in T^{-1}U<br />

    CB
    Is that all that needs to be shown or do I have to further explain this?

    Thank you
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