some facts about injective and surjective linear maps


Let k be a field and V,W be vector spacesMathworldPlanetmath over k.

PropositionPlanetmathPlanetmath. Let f:VW be an injectivePlanetmathPlanetmath linear map. Then there exists a (surjectivePlanetmathPlanetmath) linear map g:WV such that gf=idV.

Proof. Of course Im(f) is a subspacePlanetmathPlanetmathPlanetmath of W so f:VIm(f) is a linear isomorphism. Let (ei)iI be a basis of Im(f) and (ej)jJ be its completion to the basis of W, i.e. (ei)iIJ is a basis of W. Define g:WV on the basis as follows:

g(ei)=f-1(ei),if iI;
g(ej)=0,if jJ.

We will show that gf=idV.

Let vV. Then

f(v)=iIαiei,

where αik (note that the indexing set is I). Thus we have

(gf)(v)=g(iIαiei)=iIαig(ei)=iIαif-1(ei)=
=f-1(iIαiei)=f-1(f(v))=v.

It is clear that the equality gf=idV implies that g is surjective.

Proposition. Let g:WV be a surjective linear map. Then there exists a (injective) linear map f:VW such that gf=idV.

Proof. Let (ei)iI be a basis of V. Since g is onto, then for any iI there exist wiW such that g(wi)=ei. Now define f:VW by the formulaMathworldPlanetmathPlanetmath

f(ei)=wi.

It is clear that gf=idV, which implies that f is injective.

If we combine these two propositions, we have the following corollary:

Corollary. There exists an injective linear map f:VW if and only if there exists a surjective linear map g:WV.

Title some facts about injective and surjective linear maps
Canonical name SomeFactsAboutInjectiveAndSurjectiveLinearMaps
Date of creation 2013-03-22 18:32:22
Last modified on 2013-03-22 18:32:22
Owner joking (16130)
Last modified by joking (16130)
Numerical id 6
Author joking (16130)
Entry type DerivationPlanetmathPlanetmath
Classification msc 15A04