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Homework Help: Matrix representation for a transformation

  1. May 8, 2008 #1
    1. The problem statement, all variables and given/known data
    Consider the linear transformation T: R2-->R2, where
    T(x1, x2, x3)= (3x2-x3, x1+4x2+x3)

    a. Find a matrix which implements this mapping.
    b. Is this transformation one-to-one? Is it onto? Explain.

    2. Relevant equations
    [T(x)]_B = ([T]_B) (x_B)


    3. The attempt at a solution
    The matrix that implements this mapping would be the representation ([T]_B). I think that (x_B) is the vector [x1, x2, x3] and that [T(x)]_B is (3x2-x3, x1+4x2+x3) relative to the {x1, x2, x3} basis. So then ([T]_B) must be the matrix:

    0 3 -1
    1 4 1

    Row-reducing this matrix to echelon form gives 2 pivots, so the transformation is onto since the system is consistent, but it is not one-to-one because the system is linearly dependent.


    Are all my thoughts correct for this problem?
     
  2. jcsd
  3. May 8, 2008 #2

    Dick

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    The matrix is correct. T:R3->R2, not R2->R2. Your conclusions are also correct. Though I would probably figure them out in a less abstract way than counting 'pivots'.
     
  4. May 9, 2008 #3
    How would you come to the conclusions? Would you first just point out that there are more columns than rows so one-to-one is not a possibility?
     
  5. May 9, 2008 #4

    HallsofIvy

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    You might, for instance, find two distinct vectors of R3 which map into the same thing.
     
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