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Math Help - Determine whether a Linear Transformation is Invertible

  1. #1
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    Determine whether a Linear Transformation is Invertible

    Hey everyone, I have a final exam tomorrow and I would greatly appreciate it if someone could help me out with this question:

    For each of the following linear transformations T, determine whether T is invertible and justify your answer.

    1. T:R^2 -> R^3 defined by T(a_1, a_2) = (a_1 - 2a_2, a_2, 3a_1 + 4a_2).
    2. T:R^3 -> R^3 defined by T(a_1, a_2) = (3a_1 - a_2, a_2, 4a_1).
    3. T:M_2x2(R) -> M_2x2(R) defined by T(a b c d) = (a+b a c c+d).

    Both of these are 2x2 square matrices so "a" corresponds to the (1,1) entry, "b" corresponds to the (1,2) entry, "c" corresponds to the (2,1) entry, and "d" corresponds to the (2,2) entry. Same with "a+b", "a", etc.

    I think these deal with isomorphisms but I'm not sure. Also, there is a theorem that says T is invertible iff T is one-to-one and onto. But how do I prove that T is one-to-one and onto?

    Thanks for you help. BTW, what happened to Latex?
    Last edited by c_323_h; May 10th 2007 at 08:53 PM.
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  2. #2
    Junior Member
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    Dec 2006
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    I solved the first one which is not inversible.
    A one to one function/transformation is also called injective, see
    Injective function - Wikipedia, the free encyclopedia
    Also, an onto function/transformatio is surjective function, seeSurjective function - Wikipedia, the free encyclopedia

    Hope this helps!
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