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Math Help - Linear operator

  1. #1
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    Linear operator

    Please can you help me to track the question below.
    Q. Let T:X-->Y be a linear operator and dim X= dim Y = n <\infty . show that range R(T)=Y if and only if T^-1 exist.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by kinkong View Post
    Please can you help me to track the question below.
    Q. Let T:X-->Y be a linear operator and dim X= dim Y = n <\infty . show that range R(T)=Y if and only if T^-1 exist.

    Hint

    \textrm{r}(T)=Y\Leftrightarrow \dim\textrm{r}(T)=n\Leftrightarrow \dim\ker(T)=0
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  3. #3
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    gee, does everything in linear algebra reduce to the rank-nullity theorem?
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by Deveno View Post
    gee, does everything in linear algebra reduce to the rank-nullity theorem?

    Everything related with that theorem.
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  5. #5
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    your hint was useful but i couldnt figure where to go from there. please can you help me out...thanks
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by kinkong View Post
    your hint was useful but i couldnt figure where to go from there. please can you help me out...thanks

    T is injective iff \ker T=\{0\} and T is surjective iff r(T)=Y . Besides, there exists T^{-1} iff T is bijective.
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