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Showing a composition is isomorphic

  • Thread starter trap101
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  • #1
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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?
 

Answers and Replies

  • #2
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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?
 
  • #3
342
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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?


Thanks. V and W are vector spaces
 

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