How to prove that End(V) is a k-algebra

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math008
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Joined: Mon Nov 06, 2017 12:08 am UTC

How to prove that End(V) is a k-algebra

Postby math008 » Sun Dec 10, 2017 12:17 pm UTC

Hello,

I'm having difficulties proving that End(V) is a k-algebra. First, I need to prove that End(V) i.e. Hom(V,V) is a k- vector space but under what operation? I think I have to prove that End(V) is a k-algebra under the composition operation but do I also have to prove that Hom(V,V) is a vector space under the composition operation?

For example: a part of the definition a vector space in my Professor's lecture notes is:

1) a(X+Y) = aX + aY for all a in K and X,Y in V.
so on and so forth. But End(V) is perhaps not a k-algebra under Addition and Multiplication...so I have to proof that the Hom(V,V) is a vector space under the composition operation?

Maybe I'm over thinking it...but nonetheless it would be great if I would work with someone through this problem.

Many thanks.

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Qaanol
The Cheshirest Catamount
Posts: 3058
Joined: Sat May 09, 2009 11:55 pm UTC

Re: How to prove that End(V) is a k-algebra

Postby Qaanol » Sun Dec 10, 2017 6:33 pm UTC

This sounds like homework.

Can you tell us what “k-algebra” means?

Can you tell us what “End(V)” means?

Can you tell us what “Hom(V, V)” means?

Can you tell us what “V” means?
wee free kings

DavidSh
Posts: 121
Joined: Thu Feb 25, 2016 6:09 pm UTC

Re: How to prove that End(V) is a k-algebra

Postby DavidSh » Sun Dec 10, 2017 7:03 pm UTC

If you need to prove that something is a vector space, you need to give serious consideration as to the underlying field.


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