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Showing a composition is isomorphic

  1. Oct 21, 2012 #1
    Prove that if S: U-->V and T: V-->W are isomorphisms, then TS (composition) is also an isomorphism.

    Idea: So my idea was since both S, T are both isomorphic that means they both have inverses S-1 and T-1. Now this is where I'm a little grey, in order to show that TS is isomorphic, is it enough for me to obtain the identity transformation "I" by multiplying the composition TS through by S-1T-1, based on the properties of linearity?

    TS = S-1T-1(TS) = S-1S = I ? or would I have to show the existence of (TS)-1? and if so how?
     
  2. jcsd
  3. Oct 21, 2012 #2
    I would just go through and show that TS satisfies the axioms. For example, [itex]TS(u_1+u_2) = T(S(u_1)+S(u_2))[/itex] and so on. BTW, what are V and W?
     
  4. Oct 21, 2012 #3


    Thanks. V and W are vector spaces
     
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