# Showing a composition is isomorphic

1. Oct 21, 2012

### 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?

2. Oct 21, 2012

### 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?

3. Oct 21, 2012

### trap101

Thanks. V and W are vector spaces