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Math Help - Kernel and range of linear transformation

  1. #1
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    Kernel and range of linear transformation

    Hi everyone,

    I need some help with the following problem:

    Let U be the vector space of 2 x 2 matrices. Let A and B be 2 x 2 matrices. Find the kernel and range of each linear transformation:

    1) T(A) = trace(A)
    2) T(A) = A + B

    Any help is greatly appreciated!
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  2. #2
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    Quote Originally Posted by hasanbalkan View Post
    Hi everyone,

    I need some help with the following problem:

    Let U be the vector space of 2 x 2 matrices. Let A and B be 2 x 2 matrices. Find the kernel and range of each linear transformation:

    1) T(A) = trace(A)
    2) T(A) = A + B

    Any help is greatly appreciated!
    Remember the Kernel of T is the subset of U which T takes to zero. That is, a matrix A is in the kernel if and only if T(A)=0.

    So for the first transformation any matrix A where T(A)=Tr(A)=0 will be in the kernel. So what is necessary about the diagonal entries of A if Tr(A)=0?

    For the range, remember the range is the subset of the codomain of T which has a pre-image in the domain. So for the first transformation the domain of T is U and the Codomain is \mathbb{R}. We must find the subset of \mathbb{R} that is "hit" by T. This means we must find all possible real numbers a for which there exists a matrix A so Tr(A)=a. All such numbers make up the range.

    Is it possible to find a matrix A so Tr(A)=a for any real number a? If I asked, find a matrix A so Tr(A)=5, could you find one?

    You should be able to apply these ideas to the other transformation as well.
    Hope this helps. :-)
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