Math Is Fun Forum

Have you left something out? This statement does not follow from just what you have stated.

PS: I found your mistake. You want T[sup]2[/sup](v)= −v for all v ∈ V.

Then

This suggests that a basis for

All you have to do is to prove it.

NB: I've just shown that the set spans W; it remains for you to prove that it's linearly independent.

(1)

Then

(2)
This is harder to prove. It is called Sylvester's inequality: http://www.artofproblemsolving.com/Foru … hp?t=43691.

So the equation has no real solutions in x (let alone integer ones).

Since the partial derivatives are continuous for all

What is C?

Shouldn't it be

It's obvious that if

Also the factors of

Combining the two observations above gives the required solution.

Hence the conjugacy class containing

Let's call the men A, B, C, and D. Their plan is as follows: A will go to Room 1, B to Room 2, C to Room 3, and D to Room 4. If a man does not see his own wife in the first room, he will then go to the room corresponding to the man whose wife he sees in the first room – e.g. if A sees B's wife in Room 1, he opens Room 2 next; if it's C's wife he sees, he goes to Room 3 next, etc.

This strategy will improve their chances of finding the right wives in two attempts or fewer from 1/16 = 6.25% to 10/24 ≈ 41.67%.

This can be proved by induction. For the inductive step, note that

Hence, for this problem,