Linear algebra/K/Homomorphism space/Introduction/Section

Definition

Let ${}K$ be a field, and let ${}V$ and ${}W$ be ${}K$-vector spaces. Then

${}\operatorname {Hom} _{K}{\left(V,W\right)}={\left\{f:V\rightarrow W\mid f{\text{ linear mapping}}\right\}}\,$

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

${}(f+g)(v):=f(v)+g(v)\,,$

and the scalar multiplication defined by

${}(\lambda f)(v):=\lambda \cdot f(v)\,.$

Due to exercise, this is indeed a ${}K$-vector space.

Example

Let ${}W$ be a ${}K$-vector space over the field ${}K$. Then the mapping

$\operatorname {Hom} _{K}{\left(K,W\right)}\longrightarrow W,\varphi \longmapsto \varphi (1),$

is an isomorphism of vector spaces, see exercise.

Lemma

Let ${}K$ be a field, and let ${}V$ and ${}W$ be

${}K$-vector spaces. Then the following hold.

1. A linear mapping

   $\varphi \colon U\longrightarrow V$

   from another vector space ${}U$ induces a linear mapping

   $\operatorname {Hom} _{K}{\left(V,W\right)}\longrightarrow \operatorname {Hom} _{K}{\left(U,W\right)},f\longmapsto f\circ \varphi .$

2. A linear mapping

   $\psi \colon W\longrightarrow T$

   to another vector space ${}T$ induces a linear mapping

   $\operatorname {Hom} _{K}{\left(V,W\right)}\longrightarrow \operatorname {Hom} _{K}{\left(V,T\right)},f\longmapsto \psi \circ f.$

Proof

See exercise.

$\Box$

Lemma

Let ${}V$ be a ${}K$-vector space together with a direct sum decomposition

${}V=U_{1}\oplus U_{2}\,.$

Let ${}W$ be another ${}K$-vector space and let

$\varphi _{1}\colon U_{1}\longrightarrow W$

and

$\varphi _{2}\colon U_{2}\longrightarrow W$

denote linear mappings. Then we get, by setting

${}\varphi (v):=\varphi _{1}(v_{1})+\varphi _{2}(v_{2})\,,$

where ${}v=v_{1}+v_{2}$ is the direct decomposition, a linear mapping

$\varphi \colon V\longrightarrow W.$

Proof

The mapping is well-defined, since the representation ${}v=v_{1}+v_{2}$ with ${}v_{1}\in U_{1}$ and ${}v_{2}\in U_{2}$ is unique. The linearity follows from

${}{\begin{aligned}\varphi (av+bv')&=\varphi _{1}{\left((av+bv')_{1}\right)}+\varphi _{2}{\left((av+bv')_{2}\right)}\\&=\varphi _{1}{\left(av_{1}+bv'_{1}\right)}+\varphi _{2}{\left(av_{2}+bv'_{2}\right)}\\&=a\varphi _{1}(v_{1})+b\varphi _{1}(v'_{1})+a\varphi _{2}(v_{2})+b\varphi _{2}(v'_{2})\\&=a\varphi _{1}(v_{1})+a\varphi _{2}(v_{2})+b\varphi _{1}(v'_{1})+b\varphi _{2}(v'_{2})\\&=a\varphi (v)+b\varphi (v')\end{aligned}}$

$\Box$

Lemma

Let ${}K$ be a field, and let ${}V$ and ${}W$ be ${}K$-vector spaces. Let

${}V=V_{1}\oplus \cdots \oplus V_{n}\,$

and

${}W=W_{1}\oplus \cdots \oplus W_{m}\,$

de direct sum decompositions and let

$p_{j}\colon W\longrightarrow W_{j}$

denote the canonical projections. Then the mapping

$\operatorname {Hom} _{K}{\left(V,W\right)}\longrightarrow \prod _{1\leq i\leq n,\,1\leq j\leq m}\operatorname {Hom} _{K}{\left(V_{i},W_{j}\right)},f\longmapsto p_{j}\circ {\left(f{|}_{V_{i}}\right)},$

is an isomorphism. If we consider ${}\operatorname {Hom} _{K}{\left(V_{i},W_{j}\right)}$ as linear subspaces of ${}\operatorname {Hom} _{K}{\left(V,W\right)}$, then we have the direct sum decomposition

${}\operatorname {Hom} _{K}{\left(V,W\right)}\cong \bigoplus _{1\leq i\leq n,\,1\leq j\leq m}\operatorname {Hom} _{K}{\left(V_{i},W_{j}\right)}\,.$

Proof

It follows directly from fact that the given mapping is linear. In order to prove injectivity, let ${}f\in \operatorname {Hom} _{K}{\left(V,W\right)}$ with ${}f\neq 0$ be given. Then there exists some ${}v\in V$ such that

${}f(v)\neq 0\,.$

Let ${}v=v_{1}+\cdots +v_{n}$ with ${}v_{i}\in V_{i}$. Then also ${}f(v_{i})\neq 0$ for some ${}i$. Therefore, ${}(f(v_{i}))_{j}\neq 0$ for some ${}j$. Hence

${}f_{ij}=p_{j}\circ {\left(f{|}_{V_{i}}\right)}\neq 0\,.$

In order to prove surjectivity, let a family of homomorphisms ${}f_{ij}\in \operatorname {Hom} _{K}{\left(V_{i},W_{j}\right)},\,1\leq i\leq n,\,1\leq j\leq m$, be given, which we consider as mappings to ${}W$. Then the

${}f_{i}=\sum _{j=1}^{m}f_{ij}\,$

are linear mappings from ${}V_{i}$ to ${}W$. This yields via fact a linear mapping ${}\sum _{i=1}^{n}f_{i}$ from ${}V$ to ${}W$, which restricts to the given mappings.

$\Box$

Theorem

Let ${}K$ denote a field and let ${}V$ and ${}W$ denote finite-dimensional ${}K$-vector spaces. Let ${}{\mathfrak {v}}=v_{1},\ldots ,v_{n}$ be a basis of ${}V$ and ${}{\mathfrak {w}}=w_{1},\ldots ,w_{m}$ be a basis of ${}W$. Then the assignment

$\operatorname {Hom} _{K}{\left(V,W\right)}\longrightarrow \operatorname {Mat} _{m\times n}(K),f\longmapsto M_{\mathfrak {w}}^{\mathfrak {v}}(f),$

is an isomorphism

of ${}K$-vector spaces.

Proof

The bijectivity was shown in fact. The additivity follows from

${}{\left(M_{\mathfrak {w}}^{\mathfrak {v}}(f+g)\right)}_{ij}=((f+g)(v_{j}))_{i}=(f(v_{j})+g(v_{j}))_{i}=f(v_{j})_{i}+g(v_{j})_{i}={\left(M_{\mathfrak {w}}^{\mathfrak {v}}(f)\right)}_{ij}+{\left(M_{\mathfrak {w}}^{\mathfrak {v}}(g)\right)}_{ij}\,,$

where the index ${}i$ denotes the ${}i$-th component with respect to the basis ${}{\mathfrak {w}}$.

$\Box$

This means that we can consider the direct sum decomposition for the one-dimensional linear subspaces ${}Kv_{i}$ or ${}Kw_{j}$ corresponding to bases and apply fact.

Corollary

Let ${}K$ denote a field, and let ${}V$ and ${}W$ denote finite-dimensional ${}K$-vector spaces with dimensions ${}n$ and ${}m$. Then

${}\dim _{K}{\left(\operatorname {Hom} _{K}{\left(V,W\right)}\right)}=nm\,.$

Proof

This follows immediately from fact.

$\Box$