(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

HI, I'm working on this:

If n>m, prove that there is a homomorphism of F^(n) onto F^(m) with a kernel W which is isomorphic to F^(n-m).

2. Relevant equations

Def: If U and V are vector spaces over F (a field) then the mapping T of U onto V is said to be a homomorphism if:

a) (u1+u2)T=u1T+u2T

b) (a u1)T=a(u1)T

If T in addition is one-to-one, we call it and isomorphism. The Kernel ot T is defined as {u in U|uT=0} where o is the identity element of the addition in V.

3. The attempt at a solution

These are my thoughts:

It seems trivial to me that there is a homomorphism from F^(n) onto F^(m) since n>m. I just don't know how to formalize that argument. I can't get a picture in my mind to write it down.

The part about the kernel of the homomorphism isomorphic to F^(n-m) also seems to be intuitively simple..the kernel will have n-m elements in it, so there's got to be an isomorphism between the kernel and F^(n-m).

Please help me clarify and formalize this.

thanks

cc

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# Algebriac Structures

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