Question

Suppose that V is finite dimensional and S, T.

Prove that ST is invertible if and only if both S and T
areinvertible.

Answer 1

See explanation and proof below.

Explanation:

For this case we want to proof the following:

"Given that V is a finite dimensional and
`S, T \in L(V)` then ST is invertible if and only if S and T are both invertible.

In order to proof this we need to use the following result :"Given a finite dimensional vector space V, for any T \in L(V,V) we have the following properties defined: "invertibility, surjectivity, injectivity".

Proof

(> statement)

For the first part of the proof we can do this. We assume
`a_1, a_2` two vectors in V. If we assume that ST is invertible and
`T(a_1) = T(a_2)` then we have this :

`ST(a_1) = ST(a_2)`

And since ST is invertible then
`a_1 = a_2` and that implies that T is invertible.

Now if we select a vector b in V , since we know that ST is invertible, and by the surjective property defined above we have that for any
`p \in V` then
`ST(p)= S(Tp)= b` and we see that
`Tp \in V` and S is surjective and by the result above is invertible.

(< statement)

Now for this part we can assume that S and T are invertible and then
`ST i_1 = ST i_2` for any two vectors
`i_1, i_2 \in V`. Since S,T are invertible and using the surjective property we have that for any vectors
`h_1, h_2 \in V, h_1 =T i_1, h_2 = Ti_2` we have that:

`S h_1 = S h_2`

And since
`ST i_1 = ST i_2` and because S satisfy the injectivity property that implies:

`h_1 = T i_1 = T i_2 = h_2` and we can conclude that
`i_1 =i_2` and the conclusion is that ST is injective and invertible for this case.

And with that we complete the proof.