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Linear algebra - Image and Kernel

  1. Feb 4, 2012 #1
    1. The problem statement, all variables and given/known data

    Let V be a 3 dim vector space over F and e_1 e_2 and e_3 be those fix basis
    The question provide us with the linear transformation T[itex]\in[/itex] L(V) such that
    T(e_1) = e_1 + e_2 - e_3
    T(e_2) = e_2 - 3e_3
    T(e_3) = -e_1 -3e_2 -2e_3

    we are ask to find the matrix of T and the basis of ker(T) and Im(T)

    2. The attempt at a solution

    I think I find the matrix right
    where the matrix of T should be
    1 0 -1
    1 1 -3
    -1 -3 -2

    but the problem is I am not sure how can i find the ker(T) and Im(T)
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 4, 2012 #2
    Few pointers:
    o The matrix you wrote down is wrong. Look for the proper way to order the coordinates of the basis vectors.
    o For Im(T): You have to find the span of vectors
    o For Ker(T): You need to solve a matrix equation
     
  4. Feb 5, 2012 #3

    Deveno

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    i'm not seeing this. it looks correct to me.
    if one uses row-reduction, one could accomplish both at the same time.
     
  5. Feb 5, 2012 #4
    Am I wrong or one should order the vectors in columns not in rows ?
    Finding the span is just row-reducing the matrix ...
     
  6. Feb 5, 2012 #5

    Deveno

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    it appears that is what has been done.

    T(e1) = e1 + e2 - e3,

    that is: T((1,0,0)T) = 1(1,0,0)T + 1(0,1,0)T + (-1)(0,0,1)T

    = (1,1,-1)T, which appears to be the first column of sincera4565's matrix.
     
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