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Show that the range of the linear operator is not all of R^3

  1. Jul 14, 2008 #1
    1. The problem statement, all variables and given/known data
    Show that the range of the linear operator defined by the equations is not all of R3, and find a vector that is not in the range


    2. Relevant equations
    w1 = x - 2y + z
    w2 = 5x - y + 3z
    w3 = 4x + y + 2z


    3. The attempt at a solution
    can I just show that it does not have an inverse and it's a singular matrix to say it's equivalent to the range is not all or R3?
     
  2. jcsd
  3. Jul 14, 2008 #2

    quasar987

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    Yes, but you will still have to find a vector that is not in the range. So just find such a vector and solve both problems at once instead.
     
  4. Jul 14, 2008 #3
    strange, my edit was not posted.

    My question is actually how to find the vector not in the domain?
     
  5. Jul 14, 2008 #4

    HallsofIvy

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    Yes, that is the whole point.

    For what values of w1, w2, w3 does
    w1 = x - 2y + z
    w2 = 5x - y + 3z
    w3 = 4x + y + 2z
    NOT have a solution.

    Just start solving for x, y, and z and see if you run into a problem (probably dividing by 0) for some values of w1, w2, and w3.
     
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