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

We have

(ST)(T−1S−1)=SS−1=I(ST)(T^{-1}S^{-1}) = SS^{-1} = I

And

(T−1S−1)(ST)=T−1T=I(T^{-1}S^{-1})(ST) = T^{-1}T = I

Therefor (ST)(ST) is invertible and (ST)−1=T−1S−1(ST)^{-1} = T^{-1}S^{-1}.