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Math Help - linear map - matrix description

  1. #1
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    linear map - matrix description

    A = \left(\begin{array}{ccc}1&{-a}&{a}\\{-1}&a&{a+2}\\1&{2a+3}&{-3a-4}\end{array}\right)

    a \in R

    A describes the linear map T: R^3 \rightarrow R^3 for the basis B = {(1,0,-1), (1,-1,0), (1,1,1)}

    dimKerT = 2

    Find a and T(x,y,z) for every (x,y,z) in R^3

    Attempt:

    First off, I don't understand how the hint dimKerT = 2 helps us. I know it means that dimImT = 1 since dimImt + dimkerT = dimR^3...

    * Anyway, I got (and i believe these calculations are correct...)

    T(1,0,-1) = (1,2,0)
    T(1,-1,0) = (2a+3,a+3,3a+3)
    T(1,1,1) = (-a-2,-4a-6, -4a-4)

    ** since B is a basis of R^3, every vector (x,y,z) can be represented by B:

    (x,y,z) = a(1,0,-1) + b(1,-1,0) +c(1,1,1) --- and after calculations I replace a,b,c and get:
    (x,y,z,) = (-x-y+2z)(1,0,-1) + (x-z)(1,-1,0) + (x+y-z)(1,1,1)

    so T(x,y,z) = (-x-y+2z)T(1,0,-1) + (x-z)T(1,-1,0) + (x+y-z)T(1,1,1)
    T(x,y,z) = (-x-y+2z)(1,2,0) + (x-z)(2a+3,a+3,3a+3) + (x+y-z)(-a-2,-4a-6, -4a-4)

    So as you can see, I probably have to find a before I get up to step **. I am sure the dimkerT = 2 hint is useful, but I don't know why. Can someone please help?

    Thanks!!!
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    a=0 or a=-1
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  3. #3
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    a=0 or a=-1
    Haha. Thanks. would you like to expand a bit? Was I going in the right direction??
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