Suppose T ∈ L(U,V) and S ∈ L(V,W) are both invertible linear maps. Prove that ST ∈ L(U,W) is invertible and that $(ST)^{-1} = T^{-1}S^{-1}$.

Question

Suppose T ∈ L(U,V) and S ∈ L(V,W) are both invertible linear maps. Prove that ST ∈ L(U,W) is invertible and that $(ST)^{-1} = T^{-1} S ^{-1}$.

I'm not sure how to solve this proof? I'm stuck now. Is this correct so far? What should I do next? And please explain in the simplest terms to someone who has no idea about math because I'm struggling with Linear Algebra II.

So far I have:

Proof. $S$ and $T$ are invertible and the inverse of $S$ is $S^{-1}$ and $T$ is $T^{-1}$. Let $I_v\in S$ and $I_w \in T$ be identity maps on $V$ and $W$. Thus, we can write: $$ (T^{-1} S^{-1}) = T^{-1}(S^{-1} S) $$

Answer 1

Let $x \in U$ and notice that $(ST)(x) = S(T(x)) $ so we have $S^{-1}(\ (ST)(x) \ ) = T(x)$, thus $$T^{-1}( \ S^{-1}(\ (ST)(x) \ ) \ ) = x$$

Therefore $$(T^{-1}S^{-1})(ST)=Id_U$$

we can conclude that $(T^{-1}S^{-1})=(ST)^{-1}$.

Answer 2

After all that has already been said, this may be of some use : $\require{AMScd}$ \begin{CD} U @<T^{-1}<< V @<S^{-1}<<W\\ \\ U @>>T> V @>S>>W \end{CD} I would like to do more, especially loops for $1_U$ and $1_W$ but I just discovered this in LaTex thanks to the advice of Arturo Magidin. Hope this will help.