Jump to content

Linear algebra/K/Homomorphism space/Introduction/Section

From Wikiversity


Let be a field, and let and be -vector spaces. Then

is called the space of homomorphisms. It is endowed with the addition defined by

and the scalar multiplication defined by

Due to exercise, this is indeed a -vector space.


Let be a -vector space over the field . Then the mapping

is an isomorphism of vector spaces, see exercise.



Let be a field, and let and be

-vector spaces. Then the following hold.
  1. A linear mapping

    from another vector space induces a linear mapping

  2. A linear mapping

    to another vector space induces a linear mapping

Proof



Let be a -vector space together with a direct sum decomposition

Let be another -vector space and let

and

denote linear mappings. Then we get, by setting

where is the direct decomposition, a linear mapping

The mapping is well-defined, since the representation with and is unique. The linearity follows from



Let be a field, and let and be -vector spaces. Let

and

de direct sum decompositions and let

denote the canonical projections. Then the mapping

is an isomorphism. If we consider as linear subspaces of , then we have the direct sum decomposition

It follows directly from fact that the given mapping is linear. In order to prove injectivity, let with be given. Then there exists some such that

Let with . Then also for some . Therefore, for some . Hence

In order to prove surjectivity, let a family of homomorphisms , be given, which we consider as mappings to . Then the

are linear mappings from to . This yields via fact a linear mapping from to , which restricts to the given mappings.



Let denote a field and let and denote finite-dimensional -vector spaces. Let be a basis of and be a basis of . Then the assignment

is an isomorphism

of -vector spaces.

The bijectivity was shown in fact. The additivity follows from

where the index denotes the -th component with respect to the basis .


This means that we can consider the direct sum decomposition for the one-dimensional linear subspaces bzw. corresponding to bases and apply fact.


Let denote a field, and let and denote finite-dimensional -vector spaces with dimensions and . Then

This follows immediately from fact.