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Linear Algebra: Adjoint

  1. Apr 1, 2014 #1
    1. The problem statement, all variables and given/known data
    Let ##V=\mathbb{C}_2## with the standard inner product. Let T be the linear transformation deined by ##T<1,0>=<1,-2>##, ##T<0,1>=<i,-1>##. Find ##T^*<x_1,x_2>##.


    2. Relevant equations



    3. The attempt at a solution
    Find the matrix of T and then take the conjugate transpose....

    This seems like an uncharacteristically easy problem and I'm wondering if I'm missing something.
     
  2. jcsd
  3. Apr 1, 2014 #2

    Fredrik

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    I don't think you have. This looks like a problem that's only meant to test if you understand the concept of "the matrix of T". You'd be surprised how many people don't.
     
  4. Apr 1, 2014 #3
    Cool thanks!

    What happened to the post from Halls of Ivy? It was an interesting read. Oh well, maybe I'll come back after I turn in my assignment and solve it the "hard" way myself.
     
  5. Apr 1, 2014 #4

    Fredrik

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    It was deleted because he gave you too much information. Here in the homework forum, we can't really show you a complete solution unless you have already posted another complete solution.
     
  6. Apr 1, 2014 #5
    My complete answer. Does it look OK?

    The method I will use is to calculate the matrix of ##T## and take the conjugate transpose to get ##T^*<x_1,x_2>## in matrix form. Let ##M_T## be the matirx of ##T## and ##M_{T^*}## be the matrix of ##T^*##. Since
    ##T<1,0>=<1,-2>=1e_1-2e_2##
    ##T<0,1>=<i,-1>=ie_1-1e_2##
    then
    ##M_T=\left (\begin{matrix} 1&i\\-2&-1\end{matrix}\right )##
    and
    ##M_{T^*}=\left (\begin{matrix} 1&-2\\-i&-1\end{matrix}\right )##


    I'll still come back later to do it the hard way because I can see something like that popping up on a test.
     
  7. Apr 2, 2014 #6

    Fredrik

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    Yes, it looks fine. You didn't post the formula ##(M_T)_{ij}=(Te_j)_i##, but since your ##M_T## is consistent with it, I assume that you were using it.

    One of your options to calculating ##M_{T^*}## as the conjugate transpose of ##M_T##, is to start with ##(M_{T^*})_{ij}=(T^*e_j)_i##, and then rewrite the right-hand side as an inner product and use the definition of the adjoint operation.

    Since the problem is asking you to find ##T^*x## for an arbitrary ##x\in\mathbb R^2##, you don't have to explicitly write down a matrix. You can just start like this:
    $$T^*x=T^*\left(\sum_j x_j e_j\right)=\sum_j x_j T^*e_j=\sum_j x_j\sum_i(T^*e_j)_i e_i,$$ were ##(T^*e_j)_i## is defined as the ith component of the vector ##T^*e_j## with respect to the ordered basis ##(e_1,e_2)##. This is of course the ij-component of the matrix of ##T^*##, but you don't have to know that to continue this calculation.
     
    Last edited: Apr 2, 2014
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