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Transformations proof

  1. Nov 24, 2004 #1
    Let S:V --> W and T:U --> V be linear transformations. Prove that
    a) if S(T) is one-to-one, then T is one-to-one
    b) if S(T) is onto, then S is onto

    This makes intuitive sense to me, since S(T) maps U to W, but I can't figure out how to go about proving this.

    I would appreciate any help at all. Thank you.
     
    Last edited: Nov 24, 2004
  2. jcsd
  3. Nov 25, 2004 #2

    arildno

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    Dearly Missed

    a) Suppose that T is not one-to-one, but S(T) is one-to one.
    Since there exist at least u1, u2 in U, so that T[u1]=T[u2]=v1,
    then we would have S(T[u1])=S[v1]=S(T[u2])=w1, i.e, we have a contradiction, since S(T) is premised to be one-to-one.
     
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