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Thread: Matrix mapping

  1. #1
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    Matrix mapping

    Hi looking over a few past papers and I've got stuck.
    Any answers greatly appreciated

    (i)Show that the map sending A to TA determines a linear isomorphism
    f: Mnxn(F) -> Mnxn(F)*

    (ii) Let E = {TA|A belogning to Mnxn(F) and A' = A} is a subset of Mnxn(F). Compute sol(E) which is a subset of Mnxn(F)
    [You may assume standard properties of solution spaces, such as the formula
    relating dim solE and dimE. You may also assume 1 + 1 =/= 0 in F.]

    where TA(B) is trace(AB), Mnxn is an nxn matrix, F is the field and A' is the transpose of A

    Thanks
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Matrix mapping

    Quote Originally Posted by bejscs View Post
    (i)Show that the map sending A to TA determines a linear isomorphism f: Mnxn(F) -> Mnxn(F)*
    For every $\displaystyle A\in \mathcal{M}_{n\times n}(F)$ we have the map $\displaystyle T_A:\mathcal{M}_{n\times n}(F)\to F$ defined by $\displaystyle T_A(B)=\textrm{tr}(AB)$ . First of all you have to prove that $\displaystyle T_A\in \mathcal{M}_{n\times n}(F)^*$ that is, $\displaystyle T_A$ is a linear form. Write $\displaystyle T_A(\lambda_1B_1+\lambda_2B_2)=\ldots$ and apply well known trace properties. Show some work for (i) and I'll give you hints for (ii) if you need them.
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