Linear Transformation/Injective/Surjective

  1. I am extremely confused when it comes to linearly transformations and am not sure I entirely understand the concept. I have the following assignment question:

    Consider the 2x3 matrix
    A=
    1 1 1
    0 1 1
    as a linear transformation from R3 to R2.
    a) Determine whether A is a injective (one-to-one) function.
    b) Determine whether A is a surjective (onto) function.

    For a) I said that we need to solve Ax=0 and the matrix then looks like:

    1 1 1 : 0
    0 1 1 : 0

    Since x3 is a free variable A cannot be injective.

    For b) I have the matrix:
    1 1 1 : *
    0 1 1 : *
    (note that it doesn't matter what * is)

    This matrix is consistent so the matrix A is surjective.

    Am I understanding this question correctly?
     
  2. jcsd
  3. AKG

    AKG 2,585
    Science Advisor
    Homework Helper

    Do you know the definitions of injective and surjective?
     
  4. The definitions we were given are:

    Injective: A linear transformation T: R^p -->R^m is injective (one to one) if and only if the equation Tx=0 has only the solution x=0.

    Surjective: If T:R^p --> R^m is linear then T is surjective if and only if the system Tx=b is consistent for all vectors b in all real numbers m.
     
  5. AKG

    AKG 2,585
    Science Advisor
    Homework Helper

    By "the system Tx = b is consistent" you mean "the equation Tx = b has a solution" i.e. "there exists x such that Tx = b"?

    For part a), find a nonzero vector x such that Ax = 0.

    For part b), given a vector b = (b1 b2)T (T denotes transpose), can you find a vector x = (x1 x2 x3)T such that Tx = b?
     
  6. Thank you for your help, I think I understand. If not, I'll be back. :biggrin: Thanks again.
     
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