[선형대수학] 좌표 벡터 (Coordinate Vector)

$\underline{Def}$

Let $\beta=\{u_{1},\cdots,u_{n}\}$ be an ordered basis for an n-dimensional vector space $V$. $\forall x \in V$, let $a_{1},\cdots,a_{n} \in \mathbb{F}$ be the unique scalars such that $\displaystyle x=\sum_{i=1}^{n} a_{i}u_{i}$; then we define the coordinate vector of $x$ by $\beta$ (relative to $\beta$), denoted by $[x]_{\beta}$, as

$$\displaystyle [x]_{\beta}=\begin{bmatrix} a_{1} \\ \vdots \\a_{n} \end{bmatrix} \in \mathbb{F}^{n}$$

 

$\underline{Rmk}$

The map $T \colon V \rightarrow \mathbb{F}^{n}$ defined by $Tx=[x]_{\beta}$ is linear, 1-1 and onto.

 

$\underline{Proof}$

Let $x,y \in V,\, c \in \mathbb{F}$.

Then $T(cx+y)=[cx+y]_{\beta},\, cT(x)+T(y)=c[x]_{\beta}+[y]_{\beta}$.

Note

$$\begin{equation}\begin{split} \displaystyle cx+y & =\sum_{i=1}^{n} [cx+y]_{\beta,i}u_{i} \\ & = c \sum_{i=1}^{n} [x]_{\beta,i}u_{i}+\sum_{i=1}^{n}[y]_{\beta,i}u_{i} \\ & = \sum_{i=1}^{n}(c[x]_{\beta,i}+[y]_{\beta,i})u_{i} \end{split}\end{equation}$$

By the uniqueness of a coordinate representation of $cx+y$, we have $[cx+y]_{\beta,i}=c[x]_{\beta,i}+[y]_{\beta,i}$ for $i=1,\cdots,n$.

Hence,

$$\begin{equation}\begin{split} \displaystyle T(cx+y) & = [cx+y]_{\beta} \\ & =c[x]_{\beta}+c[y]_{\beta} \\ & = cT(x)+T(y) \end{split}\end{equation}$$

Suppose $x \in N(T)$; then $[x]_{\beta}=\begin{bmatrix} 0 \\ \vdots \\ 0 \end{bmatrix} \in \mathbb{F}^{n}$, so that $x=\sum_{i=1}^{n}[x]_{\beta,i}u_{i}=0$.

Thus, $N(T)=\{0\}$, that is, $T$ is 1-1 by thm.

Since $dim(V)=n=dim(\mathbb{F}^{n})$, $T$ is also onto by thm.

 

$\underline{Ex}$

With the standard ordered basis $\beta=\{1,x,x^{2}\}$ for $P_{2}(\mathbb{R})$, if $f(x)=4+6x-7x^{2}$, then $[f]_{\beta}=\begin{bmatrix} 4 \\ 6 \\ -7 \end{bmatrix}$.