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 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.
Answer:
Ann went 21 miles in the taxi
Step-by-step explanation:
-1 beacuse its closer to the postives
2/15 b/c it equals 4/30 and 1/10 equals 3/30