Linear Algebra Done Right · Problem 3D.2

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · invertible-linear-maps

Problem

Suppose

T \in \mathcal{L}(U, V)

and

S \in \mathcal{L}(V, W)

are both invertible linear maps. Prove that

ST \in \mathcal{L}(U, W)

is invertible and that

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