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:
722.24 cm
Step-by-step explanation:
The volume formula is L X W X H = V so using that information we can do 18.5 times 32 times 12.2 which gives us 722.24 cm.
Answer:
Step-by-step explanation:Re order the terms . R×2=28.32.
Divide both sides by 2 . 2r=28.32.
R=14.16.
Answer:
(-1,-1)
thats where they intersect
and thats the answer
Step-by-step explanation:
plz give brainlyest
1 Bag pinto beans cost - 4.59
3/8 of its cost ?
-> 3 /8 * 4.59
-> (3 * 4.59) / 8
-> ( 13.77)/8
= 1.72
so the 3/8 of a bag cost is $1.72