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Math Help - Linear Transformation Example

  1. #1
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    Question Linear Transformation Example

    Let T be a linear operator one the finite-dimensional space V. Suppose there is a linear operator U on V such that TU =  I. Prove that T is invertible and U =T^-1 Give an example which shows that this is false when V is not finite-dimensional.

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  2. #2
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    Quote Originally Posted by osodud View Post
    Let T be a linear operator one the finite-dimensional space V. Suppose there is a linear operator U on V such that TU =  I. Prove that T is invertible and U =T^-1 Give an example which shows that this is false when V is not finite-dimensional.

    THANKS
    TU(v) = T(U(v)) = v

    Hence range of T is entire V i.e. T is onto V.
    As T is onto AND V is finite dimensional => T is invertible.

    TU = I
    (T^-1)TU = T^-1
    U=T^-1


    As for the example in infinite case - can't think. But I guess there was something like
    T(q(x)) = q'(x)
    U(q(x) = Integral from 1 to x q(x)
    Please check it though

    Here V is F[x] - all polynomials in x
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  3. #3
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    Quote Originally Posted by aman_cc View Post
    TU(v) = T(U(v)) = v

    Hence range of T is entire V i.e. T is onto V.
    As T is onto AND V is finite dimensional => T is invertible.

    TU = I
    (T^-1)TU = T^-1
    U=T^-1


    As for the example in infinite case - can't think. But I guess there was something like
    T(q(x)) = q'(x)
    U(q(x) = Integral from 1 to x q(x)
    Please check it though

    Here V is F[x] - all polynomials in x


    Counterexample in the infinite dimensional case: let V be the real linear space of all infinite sequences and let T be the linear map defined by:

    T({a_1, a_2,....,a_n,...}):= {a_2, a_3,....}

    If U is defined as U({a_1, a_2,...}):= {0, a_1, a_2,...}, then we get
    TU = I , but T isn't invertible since it is very not 1-1 (can you see it?)

    Now just check both T, U are certainly linear maps.

    Remark: if you prefer to make the example slightly fancier you may even choose V to be the l. s. of all convergent sequences or even all the convergent to zero sequences.

    Tonio
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