7.1 선형변환(linear transformation)

예제 7.1 다음 세 개 벡터를 고려하자.

     \(\vec{x_1}=\begin{bmatrix}1\\0\\-1\\6\end{bmatrix}\), \(\vec{x_2}=\begin{bmatrix}-2\\11\\3\\5\end{bmatrix}\), \(\vec{x_3}=\begin{bmatrix}-0\\1\\2\\7\end{bmatrix}\).

     \(2\vec{x_1}+3\vec{x_2}-4\vec{x_3}=2\begin{bmatrix}1\\0\\-1\\6\end{bmatrix}+3\begin{bmatrix}-2\\11\\3\\5\end{bmatrix}-4\begin{bmatrix}0\\1\\2\\7\end{bmatrix}=\begin{bmatrix}\vec{x_1}&\vec{x_2}&\vec{x_3}\end{bmatrix}\begin{bmatrix}2\\3\\-4\end{bmatrix}=X\vec{a}\), 여기서 \(X=\begin{bmatrix}\vec{x_1}&\vec{x_2}&\vec{x_3}\end{bmatrix}\), \(\vec{a}=\begin{bmatrix}2\\3\\-4\end{bmatrix}\).

  • 선형변환 \(\vec{b}=X\vec{a}\), 여기서 \(\vec{b}=\begin{bmatrix}b_1\\b_2\\b_3\end{bmatrix}\), \(X=\begin{bmatrix}x_{11}&x_{12}&x_{13}\\x_{21}&x_{22}&x_{23}\\x_{31}&x_{32}&x_{33}\end{bmatrix}\), \(\vec{a}=\begin{bmatrix}a_1\\a_2\\a_3\end{bmatrix}\). \(X\vec{a}=\begin{bmatrix}\vec{x_1}&\vec{x_2}&\vec{x_3}\end{bmatrix}\begin{bmatrix}a_1\\a_2\\a_3\end{bmatrix}=a_1\vec{x_1}+a_2\vec{x_2}+a_3\vec{x_3}\). 단, \(\vec{x_1}, \vec{x_2}, \vec{x_3}\)는 \(X\) 의 열 벡터. 한편, \(\vec{b}^t=\vec{a}^tX^t=\begin{bmatrix}a_1&a_2&a_3\end{bmatrix}\begin{bmatrix}\vec{x_1}^t\\\vec{x_2}^t\\\vec{x_3}^t\end{bmatrix}=a_1\vec{x_1}^t+a_2\vec{x_2}^t+a_3\vec{x_3}^t\).

7.2 선형종속인 벡터

예제 7.2. 다음 벡터들을 생각해 보자.

     \(\vec{x_1}=\begin{bmatrix}3\\-6\\9\end{bmatrix}\), \(\vec{x_2}=\begin{bmatrix}0\\5\\-5\end{bmatrix}\), \(\vec{x_3}=\begin{bmatrix}2\\1\\1\end{bmatrix}\), \(\vec{x_4}=\begin{bmatrix}-6\\12\\-18\end{bmatrix}\), \(\vec{x_5}=\begin{bmatrix}2\\-3\\3\end{bmatrix}\)

(2.1) \(2\vec{x_1}+\vec{x_4}=\vec{0}\)
(2.2) \(2\vec{x_1}+3\vec{x_2}-3\vec{x_3}=\vec{0}\)
(2.3) \(a_1\vec{x_1}+a_2\vec{x_2}=\vec{0}\) 은 \(a_1=a_2=0\)일 때만 성립
(x1 <- c(3, -6, 9))
## [1]  3 -6  9
(x2 <- c(0, 5, -5))
## [1]  0  5 -5
(x3 <- c(2, 1, 1))
## [1] 2 1 1
(x4 <- c(-6, 12, -18))
## [1]  -6  12 -18
(x5 <- c(2, -3, 3))
## [1]  2 -3  3
(X123 <- cbind(x1, x2, x3))
##      x1 x2 x3
## [1,]  3  0  2
## [2,] -6  5  1
## [3,]  9 -5  1
det(X123)
## [1] -9.992007e-15
round(det(X123), digits=2)
## [1] 0
(X125<-cbind(x1, x2, x5))
##      x1 x2 x5
## [1,]  3  0  2
## [2,] -6  5 -3
## [3,]  9 -5  3
det(X125)
## [1] -30

자료 저장

save.image("chapter_7_class.rda")