Respuesta :
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 [tex] S, T \in L(V)[/tex] 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 [tex] a_1, a_2[/tex] two vectors in V. If we assume that ST is invertible and [tex] T(a_1) = T(a_2)[/tex] then we have this :
[tex] ST(a_1) = ST(a_2)[/tex]
And since ST is invertible then [tex] a_1 = a_2[/tex] 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 [tex] p \in V[/tex] then [tex] ST(p)= S(Tp)= b[/tex] and we see that [tex] Tp \in V[/tex] 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 [tex] ST i_1 = ST i_2[/tex] for any two vectors [tex] i_1, i_2 \in V[/tex]. Since S,T are invertible and using the surjective property we have that for any vectors [tex] h_1, h_2 \in V, h_1 =T i_1, h_2 = Ti_2[/tex] we have that:
[tex] S h_1 = S h_2[/tex]
And since [tex] ST i_1 = ST i_2[/tex] and because S satisfy the injectivity property that implies:
[tex] h_1 = T i_1 = T i_2 = h_2[/tex] and we can conclude that [tex] i_1 =i_2[/tex] and the conclusion is that ST is injective and invertible for this case.
And with that we complete the proof.
