Characterization of Linear Homomorphisms In Terms of Bases
Point of post: In this post we give a characterization of linear homomorphisms which has to do entirely with how the homomorphism acts on a basis.
Often it is laborious to check that that a linear transformation is injective or surjective. It turns out that there is a nice characterization of injectivity and surjectivity for finite dimensional spaces which depends entirely on how the the transformation acts on a basis of the domain space.
Theorem: Let and be finite dimensional -spaces and let . Then for any basis of it’s true that:
: First suppose that is a monomorphism. We claim that is linearly independent. To see this suppose that
but since is a monomorphism we know from prior discussion that so that the above implies that
and since is linearly independent it follows that , from where the linear independence of follows. But, this implies that may be extended to a basis as desired.
Conversely, suppose that can be injected into a basis, clearly then may be extended to a basis for . Then, we see that is linearly independent. Assume then that then
and thus by assumption , thus and so is a monomorphism.
: Suppose first that is an epimorphism. We note then that
and thus spans . But, we may then eliminate elements of to assure that the resulting set is linearly independent and still spans, namely we may find a subset of which is a basis for .
Conversely, suppose that contains a subset which spans . Then, we see that
from where it follows that is an epimorphism.
: First suppose that . Then, since is a monomorphism we have from that is linearly independent and since is an epimorphism we have from that . It easily follows that is a basis for .
Conversely, suppose that is a basis. Since may be extended to a basis (it already is one) we see from that is a monomorphism. Also, since contains a subset which is a basis for (namely itself) we see from that is an epimorphism. Thus, putting these two together implies that .
Notice that the methodology in the above proof serves to prove the following equivalent statement:
Corollary: Let and be finite dimensional spaces. Then if we see that
1. Halmos, Paul R. Finite-dimensional Vector Spaces,. New York: Springer-Verlag, 1974. Print