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Linear transformaton and inverse

  1. Dec 3, 2011 #1
    1. The problem statement, all variables and given/known data

    If T : V → W is an injective linear transformation, then T^-1: V →W is a linear transformation.

    3. The attempt at a solution

    Let w1, w2 in W. If w1=T(v1) and w2=T(v2), v1=/=v2 in V. Thus, T^-1: V →W is a function. Then, v1+v2=T^-1(w1) + T^-1(w2) and for a in F, T^-1(w1) = aT^-1(w1) = av1 for w in W.
     
  2. jcsd
  3. Dec 3, 2011 #2

    micromass

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    This is not true. Injectivity is not enough. You need bijectivity.
     
  4. Dec 3, 2011 #3
    Oh sorry, I left out a part. T:V -> W is a linear transformation, T(V) is a subspace of W.
     
  5. Dec 3, 2011 #4
    but doesn't he have to use injectivity and surjectivity to show bijectivity?
     
  6. Dec 3, 2011 #5

    micromass

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    Not enough. You need T to be surjective.

    Yes, but he didn't state that T is surjective.
     
  7. Dec 4, 2011 #6

    Deveno

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    what IS true, is that if T:V→W is an injective linear transformation, then

    T-1:T(V)→V is a linear transformation.

    now, prove this by explicitly defining what T-1 has to be.
     
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