Question

Suppose that V is finite dimensional and S, T.

Answer 1

Answer:

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 $S, T \in L(V)$ 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.