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Finding a basis for the Kernel of T

  1. Oct 10, 2011 #1
    1. The problem statement, all variables and given/known data

    So the question is a map T: R^2x2 ---> R^2x2 by T(A) = BAB, where B = (1 1)
    (1 1)

    so i made A = (a c) and T(A) = ((a+b) + (c+d) (a+b) + (c+d))
    (b d) ((a+b) + (c+d) (a+b) + (c+d))

    now it asks Find a basis for the kernel of T and compute the dimension of the kernel T.



    2. Relevant equations



    3. The attempt at a solution

    This is what I have, but im not quite sure its right.

    ker(T) = {VεR^2x2 : T (V) = 0(vector) R^2x2}

    = {(a c) : [(a+b) + (c+d) (a+b) + (c+d)] = [0 0] }
    {(b d) [a+b) + (c+d) (a+b) + (c+d)] [0 0] }

    then

    a = -b -c -d
    b = -a-c-d
    c = -a-b-d
    d = -a-b-c

    therefor the Ker(T) = {(a c) : a, b, c, d ε R^2x2}
    {(b d) }
     
    Last edited by a moderator: Oct 10, 2011
  2. jcsd
  3. Oct 10, 2011 #2

    Mark44

    Staff: Mentor

    Isn't this the same as saying all of the 2 x 2 matrices whose entries add to 0?
    There's a more systematic way to do this.

    a = -b -c -d
    b = b (obviously)
    c = .... c (ditto)
    d = ........ d (ditto)

    If you stare at the right side above, you should be able to see three vectors, many of whose entries are zero.
     
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