[선형대수학] 벡터 공간 간의 사상을 모은 집합은 벡터 공간이 된다.

$\underline{Def}$

Let $V,W$ be vector space over $\mathbb{F}$ and let $a \in \mathbb{F}$.

Let $T,U \colon V \rightarrow W$ be any maps.

We define two maps $T+U,aT \colon V \rightarrow W$ by

$$(T+U)(v)=T(v)+U(v)$$

$$(aT)(v)=aT(v)\,for\, v \in V$$

 

$\underline{Rmk}$

Let $\mathcal{F}(V,W)$ be the set of all maps from $V$ into $W$. With the operations of additaion & scalar multiplication above, $\mathcal{F}(V,W)$ is a vector space over $\mathbb{F}$. Its zero vector is the zero map $T_{0} \colon V \rightarrow W$.

 

$\underline{Def}$

Let $V,W$ be vector spaces over $\mathbb{F}$. Let $\mathcal{L}(V,W)$ denote the set of all linear maps from $V$ to $W$.

(Thus, $\mathcal{L}(V,W) \subset \mathcal{F}(V,W)$)

 

$\underline{Thm}$

$\mathcal{L}(V,W)$ is a subspace of $\mathcal{F}(V,W)$, that is, $\mathcal{L}(V,W)$ is a vector space over $\mathbb{F}$ itself.

 

$\underline{Rmk}$

Let $V,W$ be finite-dimensional vector spaces over $\mathbb{F}$ with ordered basis $\beta,\gamma$, respectively. Let $n=\vert n \vert,m=\vert \gamma \vert$.

 

$$\mathcal{L}(V,W)\;\;\;\;\;\;\; M_{m \times n}(\mathbb{F})$$

$$T \longleftrightarrow [T]_{\beta}^{\gamma}$$

$$addition \longleftrightarrow addition$$

$$scalar\, multiplication \longleftrightarrow scalar\, multiplication$$