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.

ik how do do this just give me a few minutes and i will give it to

i belive those are the ones that are under 90 degreese.