The key is to prove:
I've a problem about kernel and endomorphism. And i'm a little bit worried 'cause i'm blocked when i've just answered the first question. But let's see...
Ok, so we have a -vector space and an endomophism of .
We have and .
So, i've prooved that we always have, for all ,
Now, the question is
Ok, now of course i've searche for a little while. But the fact is : i can't see how to start the proof. I think of several way : to prove it directly, maybe by induction, i've even think of trying it with contraposition !Show that, if , then
Well, so i really don't want the answer ! Please !
But just the smallest hint ( such as, the way to prove it ) which can make me see how to keep going.
Thanks for reading me ! And i apologize if my english is poor
Ok, now i really feel stupid since i've been found blocked again a few question further :P
Here it is :
Of course, i've tried to begin, as follow :Let (assuming they exist)
Proove that, if , then
We already know that (since ).
So we just have to show that .
From now on, i'm not quite sure. We may expand as and then using the fact that .
But it doesn't seems to end up very well :/
So, i just want a small hint to help me progress, not the whole answer !
Thanks again for reading and, eventually, helping me.
Well, i've meditated on that message, but i am not quite sure i've really understood.
Ok, so i try to sum all up.
We have prooved that :
But what you just changed is the name of the variable, so we may write, for example
and i still don't see where it leads us, for we just change some names :/
Moreover, we won't be able to get back before with such equalities because .
So, i apologize, but i did not understand very well what you meant.
Ok, now i'm disturbed.
I'm afraid to say something stupid, but i think there have been a misunderstanding. :/
Because what you make me proof here is the answer to the first question i ask, nope ?
Your last post prooved what i've ask in my very first post :
didn't it ? :/Show that, if , then
Now i'm kind of embarrassed because we've gotta prove it for a integer so this induction won't work...
Hoping i didn't say something completely stupid :s
Ok, now i think i got it.
Maybe it is the way my question is asked which is weird, because it is exactly :
"Proove that, if , then . Then "
So, i've tried something desperate with a lot of complicated things and ended up with... well nothing.
But i think now it is clear.
My excuses for the misunderstanding and my thanking for the help