[선형대수학] 선형 사상을 모은 집합과 행렬을 모은 집합 간에는 선형 사상이 존재한다. (Existence of Linear Map Between Set of Linear Maps and Set of Matrices)

$\underline{Thm}$

Let $V,W$ be finite dimensional vector space over $\mathbb{F}$ with ordered basis $\beta,\gamma$, respectively.

Let $T,U \colon V \rightarrow W$ be linear & $a \in \mathbb{F}$.

Then

$$[T+U]_{\beta}^{\gamma}=[T]_{\beta}^{\gamma}+[U]_{\beta}^{\gamma},$$

$$[aT]_{\beta}^{\gamma}=a[T]_{\beta}^{\gamma}.$$

That is, the map $\mathcal{J} \colon \mathcal{L}(V,W) \rightarrow M_{m \times n}(\mathbb{F})$ with $n=\vert \beta \vert$, $m=\vert \gamma \vert$ is linear. $(T \mapsto \mathcal{J}(T)=[T]_{\beta}^{\gamma})$

 

$\underline{Proof}$

Let $\beta= \{v_{1}, \cdots, v_{n} \}, \gamma=\{ w_{1}, \cdots, w_{m} \}$.

Then

$$\displaystyle \exists ! a_{ij},b_{ij} \in \mathbb{F} \,s.t.\, T(v_{j})=\sum_{i=1}^{n} a_{ij}w_{i}, U(v_{j})=\sum_{i=1}^{m} b_{ij}w_{i},\, for\, 1 \leq i \leq m,\, 1 \leq j \leq n$$

So

$$\displaystyle (T+U)(v_{j})=\sum_{i=1}^{m} (a_{ij}+b_{ij})w_{i}$$

$$\displaystyle (aT)(v_{j})=\sum_{i=1}^{m} (a  a_{ij})w_{i}\, for \, 1 \leq j \leq n$$

Thus,

$$[T+U]_{\beta}^{\gamma}=(a_{ij}+b_{ij})=(a_{ij})+(b_{ij})=[T]_{\beta}^{\gamma}+[U]_{\beta}^{\gamma},$$

$$[aT]_{\beta}^{\gamma}=(a  a_{ij})=a(a_{ij})=a[T]_{\beta}^{\gamma}$$

 

$\underline{Ex}$

$$T, U \colon \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$$

$$T(a_{1}, a_{2})=(a_{1}+3a_{2},0,2a_{1}-4a_{2}),\, U(a_{1}, a_{2})=(a_{1}-a_{2},2a_{1},3a_{1}+2a_{2})$$

$$\beta=\{ e_{1}, e_{2} \}, \gamma=\{ e_{1}, e_{2}, e_{3} \}$$

$$(T+U)(a_{1},a_{2})=(2a_{1}+2a_{2},2a_{1},5a_{1}-2a_{2})$$

$$[T]_{\beta}^{\gamma}=\begin{bmatrix} 1 & 3 \\ 0 & 0 \\ 2 & -4 \end{bmatrix}, [U]=_{\beta}^{\gamma}=\begin{bmatrix} 1 & -1 \\ 2 & 0 \\ 3 & 2 \end{bmatrix},[T+U]_{\beta}^{\gamma}=\begin{bmatrix} 2 & 2 \\ 2 & 0 \\ 5 & -2 \end{bmatrix}$$