- #1
hh13
Field of modulo p equiv classes, how injective linear map --> surjectivity
Let F_{p} be the field of modulo p equivalence classes on Z. Recall that |F_{p}| = p.
Let L: F_{p}^{n}-->F_{p}^{n} be a linear map. Prove that L is injective if and only if L is surjective.
|F_{p}^{n}| = p^{n}. We also know that for an F-linear map L: V-->V, that L is onto iff {v_{1},...,v_{m}} spans V, given that DimV=m<infinity and that {v_{1},...,v_{m}} are column vectors of the mxm matrix associated with L.
And that L is injective iff {v_{1},...,v_{m}} is linearly independent.
And that L is bijective iff {v_{1},...,v_{m}} is a basis.
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 F_{p}^{n} such that L(v)[tex]\neq[/tex] v' for every v in F_{p}^{n}. 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 =)
Homework Statement
Let F_{p} be the field of modulo p equivalence classes on Z. Recall that |F_{p}| = p.
Let L: F_{p}^{n}-->F_{p}^{n} be a linear map. Prove that L is injective if and only if L is surjective.
Homework Equations
|F_{p}^{n}| = p^{n}. We also know that for an F-linear map L: V-->V, that L is onto iff {v_{1},...,v_{m}} spans V, given that DimV=m<infinity and that {v_{1},...,v_{m}} are column vectors of the mxm matrix associated with L.
And that L is injective iff {v_{1},...,v_{m}} is linearly independent.
And that L is bijective iff {v_{1},...,v_{m}} is a basis.
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 F_{p}^{n} such that L(v)[tex]\neq[/tex] v' for every v in F_{p}^{n}. 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 =)