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.
0-1 I think because if they are all equal you cannot make another one
Answer:
Step-by-step explanation:
1
Answer: Brain paid $ 45.5 for the jacket.
Step-by-step explanation:
Given: Original price of jacket = $ 65
Discount -= 30%
Price after discount = Original price - 30% of Original price
= 65 - 30% of 65
= 65 - (0.3) x 65 [30% = 0.3]
= (65) (1-0.3 ) [Taking 65 common]
= (65)(0.7)
= $ 45.5
Hence, Brain paid $ 45.5 for the jacket.
Answer:
B) $28.25
Step-by-step explanation:
To solve this problem, you need to find how much the discount is.
To find the discount you multiply 75% by $112.99, which equals $84.74.
Now you have to find how much the bike will cost with the discount.
So you have to subtract $112.99 by $84.74, which equals $28.25
So the discounted price of the bike is $28.25.