Suppose that v is finite dimensional and s, t.
prove that st is invertible if and only if both s and t
areinvertible.

See explanation and proof below.

Step-by-step explanation:

For this case we want to proof the following:

"Given that V is a finite dimensional and 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 two vectors in V. If we assume that ST is invertible and then we have this :

And since ST is invertible then 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 then and we see that 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 for any two vectors . Since S,T are invertible and using the surjective property we have that for any vectors we have that:

And since and because S satisfy the injectivity property that implies:

and we can conclude that and the conclusion is that ST is injective and invertible for this case.

And with that we complete the proof.

step-by-step explanation:

try remembering the steps