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Math Help - Trace(A) - Finding Nullspace and Range; Deciding if Onto and/or One-to-One

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    Trace(A) - Finding Nullspace and Range; Deciding if Onto and/or One-to-One

    See attachments for problem and note that I want to solve the problem for n=2 only. Also, my attempted solution is attached as well.

    Question:

    N(T) denotes the nullspace of T. I found the nullspace, but I'm not sure if I have the correct basis for it or if I have written the solution space with the correct notation. Obviously, the nullspace will be the set of vectors where the 2 diagonal components of the matrix are additive inverses of one another. But given that either of the diagonal components could be negative, how do I write this in proper format?


    Also, did I get the correct basis?

    Thanks for your time.
    Attached Thumbnails Attached Thumbnails Trace(A) - Finding Nullspace and Range; Deciding if Onto and/or One-to-One-la3.jpg   Trace(A) - Finding Nullspace and Range; Deciding if Onto and/or One-to-One-la2.jpg   Trace(A) - Finding Nullspace and Range; Deciding if Onto and/or One-to-One-la.jpg  
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    MHF Contributor FernandoRevilla's Avatar
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    Re: Trace(A) - Finding Nullspace and Range; Deciding if Onto and/or One-to-One

    We have T:M_{n\times n}(F)\to F given by T(A)=\textrm{tr}(A) . For n=2 , A=\begin{bmatrix}{x}&{y}\\{z}&{t}\end{bmatrix}\in N(T) iff x+t=0 so, A\in N(T) iff A has the form:



    with x,y,z\in F. Easily proved, those three matrices are linearly independent and as a consequence form a basis for N(T) .
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