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Basis of range of a matrix relative to some bases

  1. Feb 24, 2010 #1
    1. The problem statement, all variables and given/known data
    Let T be the linear transformation of R5 into R3 that has the matrix

    A =
    1 3 2 0 -1
    2 6 4 6 4
    1 3 2 2 1

    relative to the bases [(1,1,1,1,1), (1,1,1,1,0), (1,1,0,0,0), (1,0,0,0,0), (0,0,0,0,1)] of R5 and [(1,1,1), (0,1,0), (1,0,0)] of R3. Find a basis for the range of T.

    2. Relevant equations

    3. The attempt at a solution

    The whole ordeal with the different bases for R5 and R3 confuses me. If this was just an ordinary matrix, I'd have row reduced and gotten my basis for the range to be
    {[1 2 1]^T and [0 6 2]^T.

    But this isn't the correct answer. The correct answer is

    [(4/3, 1, 1), (1/3, 1, 0)].

    Any suggestions? Much thanks.
  2. jcsd
  3. Feb 25, 2010 #2


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    The vectors you found are representations relative to the basis {(1,1,1), (0,1,0), (1,0,0)}. The answer you gave is relative to the natural basis. You just need to change representations from one basis to the other. Keep in mind that a basis is not unique, so you may come up with two vectors that don't match the answer, but they just need to have the same span.
  4. Feb 25, 2010 #3
    Hi, thank you very much for your answer.
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