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Nullity Question

  1. Oct 10, 2008 #1

    _F_

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    1. The problem statement, all variables and given/known data
    Find the nullity of the linear transformation T:M_2x3(F) -> M_2x2(F) defined by:

    T([a11, a12, a13; a21, a22, a23]) = ([2*a11 - a12, a13 + 2*a12; 0, 0])


    3. The attempt at a solution
    N(T) = {x in M_2x3(F) | T(x) = 0}

    2*a11 - a12 = 0 => a11 = (a12)/2
    a13 + 2*12 = 0 a13 = -2*a12

    N(T) = {[a/2, a, -2*a; 0, 0, 0] | a in F}

    Basis(N(T)) = {[1/2 1 -2; 0, 0, 0]}

    nullity(T) = 6

    But nullity(T) should be 4...

    Any help is appreciated.
     
  2. jcsd
  3. Oct 10, 2008 #2

    Defennder

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    Why are these entries 0's? Look at the formula given for T. Do a21,a22,a23 appear anywhere in the transformed 2x2 matrix?
     
  4. Oct 11, 2008 #3

    _F_

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    You're right. But that said, if you take away the zero's you still get a 1x3 matrix whose dimension is 3...
     
  5. Oct 11, 2008 #4

    Defennder

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    You meant I a 2x3 matrix I presume. And it doesn't make sense to talk about the dimension of a matrix. You meant the dimension of the nullspace. And it isn't 3. How did you arrive at that?
     
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