[응용선형대수] Linear Transformations

matrix의 처음 등장은 Linear equation을 matrix의 곱으로서 표현한다는 것에서 시작되었습니다.

그러나 matrix를 다른 관점에서도 이해할 수가 있습니다.

 

바로 matrix를 Linear transformations으로 해석하는 것입니다.

 

$A : m\times n$

$A: \mathbb{R}^{n}\rightarrow \mathbb{R}^{m}$

 

$\text{ex) }$

 

 

1) 벡터를 스칼라 c배 하는 것

$A=\begin{bmatrix} c&0\\0&c \end{bmatrix}$

 

$A\begin{bmatrix} x\\y\end{bmatrix}=\begin{bmatrix}cx\\ cy\end{bmatrix}$

 

 

2) 벡터를 $y=x$ 대칭

$A=\begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix}$

 

$A\begin{bmatrix}x\\y \end{bmatrix}=\begin{bmatrix}y\\x\end{bmatrix}$

 

3) 벡터를 x축에 정사영

$A=\begin{bmatrix}1&0\\0&0\end{bmatrix}$

 

$A\begin{bmatrix}x\\y \end{bmatrix}=\begin{bmatrix}x\\0 \end{bmatrix}$

 

 

 

 

$\text{Definition) }$

$T:V\rightarrow W \text{ is called Linear Transformation (V,W : vector spaces)}$

$\text{if }T(ax+by)=aT(x)+bT(y)$

$a,b\in \mathbb{R}$

$x,y \in V$

 

$\text{ex) }A: \mathbb{R}^{2}\rightarrow \mathbb{R}^{3}$

$A\begin{bmatrix}1\\ 0\end{bmatrix}=\begin{bmatrix}1\\ 2\\3\end{bmatrix}$

$A\begin{bmatrix}0\\1\end{bmatrix}=\begin{bmatrix}4\\ 5\\6\end{bmatrix}$

 

$T: V \rightarrow W$

$v_{1},v_{2},\ldots,v_{n}\text{ form a basis of V}$

$\text{If we know }Tv_{i}(i=1,\ldots,n)\text{ then we can determine }Tv$

$v=c_{1}v_{1}+c_{2}v_{2}+\ldots+c_{n}v_{n}$

$Tv=c_{1}Tv_{1}+\ldots+c_{n}Tv_{n}$

 

$\text{Question) Does any linear transformation lead to a matrix?}$

 

$\text{ex) }$

 

$\text{1) Polynomials}$

 

$P_{n}=\{\text{Polynomials with deg }\geq n\}\text{ is a vector space}$

$P(t)=a_{0}+a_{1}t+a_{2}t^{2}+\cdots+a_{n}t^{n}$

$\text{basis : }1,t,t^{2},\cdots, t^{n}\left\{\begin{matrix}\text{Span }P_{n}\\ \text{Linearly independent}\end{matrix}\right.$

 

$c_{0}+c_{1}t+\cdots+c_{n}t^{n}=0 \Leftrightarrow c_{0}=c_{1}=\cdots=c_{n}=0$

 

$\text{2) Differential}$

 

$\frac{d}{dt}P_{n} \text{ : Linear}$

$\frac{d}{dt}(aP_{1}(t)+bP_{2}(t))=a\frac{dP_{1}}{dt}+b\frac{dP_{2}}{dt}$

 

$\text{basis : }1,t,t^{2},t^{3}$

각 basis를 $e_{1},e_{2},e_{3},e_{4} \in \mathbb{R}^{4}$

 

$a_{0}+a_{1}t+a_{2}t^{2}+a_{3}t^{3}\leftrightarrow \begin{bmatrix}a_{0}\\ a_{1}\\ a_{2}\\ a_{3}\end{bmatrix}\in \mathbb{R}^{4}$

 

$\frac{d}{dt}1(e_{0})=0$

$\frac{d}{dt}t=1=e_{1}$

$\frac{d}{dt}t^{2}=2t=2e_{2}$

$\frac{d}{dt}t^{3}=3t^{2}=3e_{3}$

 

$1+t+t^{2} \overset{\text{diff}}{\longrightarrow} 1+2t$

이를 matrix로 표현하면 다음과 같다.

 

$\text{differential transformation matrix : }A_{\text{diff}}=\begin{bmatrix}0 & 1 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 3\\ 0 & 0 & 0 & 0\end{bmatrix}\begin{bmatrix}1\\ 1\\ 1\\ 0\end{bmatrix}=\begin{bmatrix}1\\ 2\\ 0\\ 0\end{bmatrix}$

 

$\text{integral transformation matrix : }A_{\text{int}}=\begin{bmatrix}0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & \frac{1}{2} & 0 & 0\\ 0 & 0 & \frac{1}{3} & 0\\ 0 & 0 & 0 & \frac{1}{4}\end{bmatrix}$

 

$A_{\text{int}}A_{\text{diff}}=\begin{bmatrix}0 & 1 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 3 & 0\\ 0 & 0 & 0 & 0 & 4\end{bmatrix}\begin{bmatrix}0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & \frac{1}{2} & 0 & 0\\ 0 & 0 & \frac{1}{3} & 0\\ 0 & 0 & 0 & \frac{1}{4}\end{bmatrix} =\begin{bmatrix}1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{bmatrix}$

 

 

+ Some examples for $2\times 2$ matrix

(1). Rotation by $\theta$

 

$Q_{\theta}\begin{bmatrix}1\\ 0\end{bmatrix}=\begin{bmatrix}\cos\theta \\ \sin\theta\end{bmatrix}$

 

$Q_{\theta}\begin{bmatrix}0\\ 1\end{bmatrix}=\begin{bmatrix}-\sin\theta \\ \cos\theta\end{bmatrix}$

 

$Q_{\theta}=\begin{bmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{bmatrix}$

 

$Q_{\theta}^{2}=Q_{\theta}\cdot Q_{\theta}=Q_{2\theta}$

 

$\begin{bmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{bmatrix} \begin{bmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{bmatrix} = \begin{bmatrix}\cos 2\theta & -\sin 2\theta \\ \sin 2\theta & \cos 2\theta \end{bmatrix}$

$\text{ex) }Q_{\theta}\cdot Q_{\varphi}=Q_{\theta+\varphi}$

 

 

 

(2). Projection on the line

 

 

$P\begin{bmatrix} 1 \\ 0\end{bmatrix}=\cos\theta \begin{bmatrix}\cos\theta \\ \sin\theta \end{bmatrix}=\begin{bmatrix}\cos^{2}\theta \\ \cos\theta \sin\theta \end{bmatrix}$

$P\begin{bmatrix} 0 \\ 1 \end{bmatrix}=\sin\theta \begin{bmatrix} \cos\theta \\ \sin\theta \end{bmatrix} = \begin{bmatrix} \cos\theta \sin\theta \\ \sin^{2}\theta \end{bmatrix}$

 

$P=\begin{bmatrix}\cos^{2}\theta & \cos\theta\sin\theta\\ \cos\theta\sin\theta & \sin^{2}\theta\end{bmatrix}$

 

$P^{2}=P$

이미 projection한 것에 대해 또 projection을 하면 처음 Projection한 것과 동일한 결과가 나옵니다.

 

 

 

(3). Reflection

 

$\frac{1}{2}\left ( H \begin{bmatrix}x\\ y\end{bmatrix} + \begin{bmatrix}x\\ y\end{bmatrix} \right )=P\begin{bmatrix}x\\ y\end{bmatrix}$

$\Rightarrow H \begin{bmatrix}x\\ y\end{bmatrix}=(2P-I)\begin{bmatrix}x\\ y\end{bmatrix}$

$H=2P-I$

$H^{2}=4P^{2}-4P+I^{2}$

 

$y=x$ 축에 대해 reflection하면 다시 reflection했을 때 ($\Rightarrow H^{2}$) 원래 자리로 돌아온다. ($\Rightarrow I$)