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Find the Kernel of the Trace of a Matrix

  1. Oct 15, 2012 #1
    1. The problem statement, all variables and given/known data

    Let F : Mnn(R) → R where F(A) =tr(A). Show that F is a linear transformation. Find the kernel of F as well as its dimension. What is the image of F?


    2. Relevant equations



    3. The attempt at a solution

    I have shown that it is a linear transformation. But I am not sure about the Ker(F) and Im(F),
    would Ker(F) just be {tr(A), for A in Mnn}? And would the Im(F) just be {a : a[itex]\in[/itex]ℝ}? Thanks.
     
  2. jcsd
  3. Oct 15, 2012 #2

    HallsofIvy

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    Do you understand what the problem is asking? You are gvien a linear transformation that maps every matrix to a number, its trace. This problem is asking for the trace of that linear transforamation- the set of matrices that are mapped to 0. It is asking for a set of matrices, not a set of numbers.
     
  4. Oct 15, 2012 #3
    Okay, so would I say kernel is {A where tr(A)=0, for A in Mnn}? So what would the image be then? Thanks
     
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