Linear Algebra: Determine if V is a Vector space

[Steven G]

The inverse is 3n - 1 for all n in the set N_0?

But you do not want to find the inverse of all elements in N_0. You want to find inverses of all elements in the set {0,1,2}

 

[A--]

But you do not want to find the inverse of all elements in N_0. You want to find inverses of all elements in the set {0,1,2}

So its 3n - 1 for all n in the set V = {0 1 2} ?

 

[Steven G]

So its 3n - 1 for all n in the set V = {0 1 2} ?

Let's see. Using your method, the inverse of 0 is 3*0-1 or -1, the inverse of 1 is 3*1-1 or 2 and the inverse of 2 is 3*2- 1 or 7. Do you think that this is correct?
Even if what you wrote is correct, you are not doing this the easy way. Just look at the table you have and see, for example, where it say 2+? =0 and then the inverse of 2 is ?

 

[A--]

Let's see. Using your method, the inverse of 0 is 3*0-1 or -1, the inverse of 1 is 3*1-1 or 2 and the inverse of 3 is 3*3- 1 or 8. Do you think that this is correct?
Even if what you wrote is correct, you are not doing this the easy way. Just look at the table you have and see, for example, where it say 2+? =0 and then the inverse of 2 is ?

Yes i know the inverse of 1 is 2 and inverse of 2 is 1 but how do I generalise this ?

 

[Steven G]

Yes i know the inverse of 1 is 2 and inverse of 2 is 1 but how do I generalise this ?

Why do you want to generalize it? What if instead of 0, 1 and 2 we had a, b and c?

 

[Steven G]

OK, how do you generalize it? There are 3 inverses. So define a quadratic equation to be f(x) = a + bx + cx^2
We know f(0)=0 , f(1) = 2 and f(2) =1. Now solve for a, b and c.

 

[A--]

OK, how do you generalize it? There are 3 inverses. So define a quadratic equation to be f(x) = a + bx + cx^2
We know f(0)=0 , f(1) = 2 and f(2) =1. Now solve for a, b and c.

when x = 0 a + 0 + 0 = 0
so a = 0
b + c = 2 when x = 1
2b + 4c = 1 when x = 2

solving simultaneous equations
7/2 x - 3/2x^2 = the inverse?

 

[Dr.Peterson]

Yes i know the inverse of 1 is 2 and inverse of 2 is 1 but how do I generalise this ?

There is nothing to generalize! The set V is {0, 1, 2}; you have found the inverse of each element (the inverse of 0 is 0), so you are done.

Previously, I said this:

Don't think about inverses in Z; think about inverses in V, which sometimes have to be worked out one by one. When you have done this, you may see a formula you could write, and then justify it; but that isn't necessary.

Answers do not have to be formulas. In this case, it happens that you could write v^-1 = 3 - v (mod 3); do you see that this gives the correct result? But that doesn't make this a better answer than just listing the inverse of each element. (It does tie the answer to modular arithmetic, which underlies the problem, but that's really just notation.)

 

[Steven G]

when x = 0 a + 0 + 0 = 0
so a = 0
b + c = 2 when x = 1
2b + 4c = 1 when x = 2

solving simultaneous equations
7/2 x - 3/2x^2 = the inverse?

Yes, but it is not necessary at all to generalize it. I told you one way to do it because you wanted the formula, but it is not necessary. To answer your question, yes the inverse of 0 is f(0), inverse of 1 is f(1) and the inverse of 2 is f(2) where f(x) = 7/2 x - 3/2x^2