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Homework Help: Kernel and image of linear transformation

  1. Oct 17, 2011 #1
    1. The problem statement, all variables and given/known data
    For the linear transformation T: R4 --> R3 defined by TA: v -->Av
    find a basis for the Kernel of TA and for the Image of of TA where A is
    2 4 6 2
    1 3 -4 1
    4 10 -2 4

    2. Relevant equations

    Let v =
    a1 b1 c1
    a2 b2 c2
    a3 b3 c3
    a4 b4 c4
    3. The attempt at a solution
    so v is a 4x3 matrix, and Ker(T) would just be the solution for Av = 0.
    I was unsure as to what the Image would be give by. Is it the matrix
    2a1+4b1+6c1+2d1, 2a2+4b2+6c2+2d2, 2a3+4b3+6c3+2d3,
    1a1+3b1-4c1+d1, ... etc

    (just the general solution of the multiplication)
    Which generalizes to
    2 0 2
    0 1 2
    so the basis is [1, 2, -1]

    How would I find a basis for the Kernel?
    Last edited: Oct 17, 2011
  2. jcsd
  3. Oct 17, 2011 #2


    Staff: Mentor

    You're really heading down the wrong path here. v is a vector in R4.
    Ker(T) is the set of all vectors v in R4 such that Tv = 0.
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