1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Algebriac Structures

  1. Jan 25, 2007 #1
    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.
  2. jcsd
  3. Jan 25, 2007 #2


    User Avatar
    Homework Helper

    what is F^n explicitly? Write down a general element in F^n and it should be obvious where to send this in F^m to get a homomorphism (usually called a linear map in this case) with the desired properties. Think about the case F=R, the real numbers.
  4. Jan 25, 2007 #3
    Without working out the details explicitly, I would imagine the multivariate projection map from [tex] F^n \rightarrow F^m [/tex] should satisfy the homomorphism conditions.

    Furthermore, have you considered using the first isomorphism theorem for rings to help formally show that [tex] F^{n-m} [/tex] is isomorphic to the kernel?

    It shouldn't be too hard to show that the projection map [tex] \pi (x) [/tex] is surjective, and so that [tex]Image(\pi(x))=F^m[/tex] is a subring of [tex]F^n[/tex]. Then [tex]F^m[/tex] is isomorphic to [tex]F^n \setminus_{Ker(\pi)}[/tex]. Consider the natural map between these two sets and then see if you can get anything from there.

    This might be a bit over the top, but was the first thing that came to my mind.
  5. Jan 27, 2007 #4
    Thanks for the input. It's been a year since I studied rings and I'm all rusty. I'm pretty sure I understand this one and I think I got it.

    My next question is this:Prove that there exists an isomorphism from [tex]F^n[/tex] into [tex]Hom(Hom(F^n , F),F).[/tex]
    Again, I'm all rusty on this stuff, so any input will be helpful.
    My confusion here lies in that I just finished proving (by contradiction) that [tex]F^1[/tex] is not isomorphic to [tex]F^n[/tex] for n>1.
    So I'm not sure what [tex]F[/tex] is or what [tex]Hom(F^n,F)[/tex] looks like. It seems to me that [tex]F^n[/tex] is finite and [tex]F[/tex] is infinite and then my mind just starts going in circles. Please help me understand this.
    Last edited: Jan 27, 2007
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Algebriac Structures
  1. Algebraic structure (Replies: 4)

  2. Group Structure (Replies: 19)

  3. Number and structure (Replies: 13)

  4. Ring Structures (Replies: 2)