Question

Suppose T ∈ ℒ(, ) and ∈ ℒ(, W) are both invertible linear

maps.
Prove
that T ∈ ℒ(, W) is invertible and that (T)−1 = T−1−1

.

Answer 1

To prove that the composition TS is invertible with an inverse of (TS)^{-1} = S^{-1}T^{-1}, we use the properties of composition and inverse of linear maps. By sequentially applying inverses S^{-1} and T^{-1} to an element in W, we find the unique preimage in V, which confirms that (TS)^{-1} = S^{-1}T^{-1} is the inverse of TS.

Step-by-step explanation:

The student's question asks to prove that if
`T ∈ ℒ(V, U)`are both invertible linear maps, then the composition is also invertible with the inverse given by
`T ∈ ℒ(V, W)` To prove this, we need to show that both composition of functions and inverse functions work as expected in the context of linear maps.

We can start by considering the composition TS, which takes an element v from V, first applies T to get T(v) in U, and then applies S to get S(T(v)) in W. Since both T and S are invertible, there exist maps T^{-1} and S^{-1} such that T^{-1}(T(v)) = v for all v in V, and S^{-1}(S(u)) = u for all u in U. Consequently, for any w in W, there exists a unique v in V such that w = S(T(v)).

To find the inverse of TS, consider an arbitrary element w in W. Applying S^{-1} first, we get u = S^{-1}(w) which is in U. Then applying T^{-1} to u, we get v = T^{-1}(u) which is back in V. Therefore, the composition
`(S^(-1)T^(-1))(w) = T^(-1)(S^(-1)(w))`in V such that TS(v) = w, proving that is indeed the inverse of
`(TS)^(-1) = S^(-1)T^(-1)`.