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February 27th, 2011, 12:47 PM   #1
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Kernel of a linear map over a vector space k

Hi

Let be a linear map over a vector space where is generated by

and where

Also, for each

Show that is non-zero.

Please give me just a hint.
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February 27th, 2011, 01:50 PM   #2
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Re: Kernel of a linear map over a vector space k

Quote:
Originally Posted by Saba
Also, for each
Would you please clarify what you mean by this ? There is no multiplication on a standard vector space, only an external law (Field, Vector space) -> Vector space. Are you working with an algebra or something ?
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February 27th, 2011, 02:00 PM   #3
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Re: Kernel of a linear map over a vector space k

is a linear map on a -vector space M with for some

is a field

is -algebra generated by satisfy the relation
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February 27th, 2011, 02:01 PM   #4
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Re: Kernel of a linear map over a vector space k

Thanx in advance
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February 27th, 2011, 03:02 PM   #5
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Re: Kernel of a linear map over a vector space k

Hi Saba,

your notaion is confusing. Infact you keep changing it. The good thing is that you don't need anything of the algebra stuff for your result: Any nilpotent endomorphism f (i.e. ) of a nonzero vector space (or basically any structure you can think of) has this property.

As a hint: Show the stronger result: The composition of any two injective morphisms is injective. Generalise to n morphisms and conclude your claim.
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February 27th, 2011, 03:08 PM   #6
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Re: Kernel of a linear map over a vector space k

if the map is injective , then ker(f) =0

But here ,show that ker f is NON-zero
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February 28th, 2011, 03:27 AM   #7
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Re: Kernel of a linear map over a vector space k

Quote:
Originally Posted by Saba
But here ,show that ker f is NON-zero
Well, you just wanted a hint. I would suggest to assume for contradiction that f is injective, then show that is injective. But this is absurd, as and your vector space is non-zero.
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