1. Limited time only! Sign up for a free 30min personal 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!

Field of modulo p equiv classes, how injective linear map -> surjectivity

  1. Nov 24, 2009 #1
    Field of modulo p equiv classes, how injective linear map --> surjectivity

    1. The problem statement, all variables and given/known data

    Let Fp be the field of modulo p equivalence classes on Z. Recall that |Fp| = p.

    Let L: Fpn-->Fpn be a linear map. Prove that L is injective if and only if L is surjective.

    2. Relevant equations

    |Fpn| = pn. We also know that for an F-linear map L: V-->V, that L is onto iff {v1,...,vm} spans V, given that DimV=m<infinity and that {v1,...,vm} are column vectors of the mxm matrix associated with L.

    And that L is injective iff {v1,...,vm} is linearly independent.

    And that L is bijective iff {v1,...,vm} is a basis.

    3. The attempt at a solution

    I just don't know what it means for a linear map to have the field of modulo p equivalence classes. on Z, let alone try and do this proof. If we suppose L is injective, then kerL=0. If L is NOT surjective, then there exists some v' in Fpn such that L(v)[tex]\neq[/tex] v' for every v in Fpn. I believe this has something to do with the field of modulo p equivalence classes, but again our professor hasn't really taught us those...

    Thanks =)
     
  2. jcsd
  3. Nov 24, 2009 #2

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Re: Field of modulo p equiv classes, how injective linear map --> surjectivity

    It's a trick question; the vector space structure is irrelevant!

    Actually, IIRC, you could use the exact same proof that you would use for finite-dimensional real vector spaces. The field of scalars is irrelevant.

    However, the fact you're working with a finite field makes things that much simpler.
     
  4. Nov 24, 2009 #3
    Re: Field of modulo p equiv classes, how injective linear map --> surjectivity

    But how do we do that?

    I'm having a hard time trying to go about proving linear maps are isomorphisms. I've heard that if, in a linear map L: V--W, if the basis for V is {v1,...,vn}, then if {L(v1),...,L(vn)} works as a basis for W, then it proves that V and W are isomorphic to each other and so L is an isomorphism...but I don't get why...or maybe I've heard wrong?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Field of modulo p equiv classes, how injective linear map -> surjectivity
  1. Equiv. Class Problem (Replies: 5)

Loading...