Suppose that W is finite dimensional and T S LW Prove that

Suppose that W is finite dimensional and T, S ? L(W). Prove that T S is invertible if and only if both T and S are invertible.

Solution

part a.

Assume that both S and T are invertible.This implies that both S and T are injective.

now we need to prove that ST is invertible.so it is sufficient to show that ST is injective.

so proof that ST is injective:

let STw = 0, so it is sufficient to show that w = 0.

Since S is injective we conclude that Tw = 0, since T is injective we conclude that w = 0.

this completes first part of the proof.

part b.

Assume that ST is invertible.we need to prove that both S and T are invertible.

Let\'s first prove that T is invertible.so prove it by contradiction.so assume T is not invertible, hence not injective.

since T is not injective, there doesn\'t exist w !=0 such that Tw = 0. this implies that S(Tw) = 0, since S(Tw) = ST(w),

we have ST(w) = 0 with a non zero w.so we conclude that ST is not injective.

again this implies that ST is not invertible which is contradiction. so T must be invertible.

Now let\'s prove that S is invertible.so it is sufficient to show that S is surjective.It is sufficient to show that for any w in W, there exist u in W such that Su = w.

let w is in W, since ST is invertible.so it is surjective also.it follows that there exists v in W such that STv = w.since

STv = S(Tv)

we put u = Tv

and obtain Su = w as required.