Showing a composition is isomorphic

[trap101]

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?

 

[Robert1986]

I would just go through and show that TS satisfies the axioms. For example, TS(u_1+u_2) = T(S(u_1)+S(u_2)) and so on. BTW, what are V and W?

 

[trap101]

I would just go through and show that TS satisfies the axioms. For example, TS(u_1+u_2) = T(S(u_1)+S(u_2)) and so on. BTW, what are V and W?

Thanks. V and W are vector spaces