부분공간의 기저로부터 확장된 기저에 대한 선형변환의 행렬표현
📂선형대수 부분공간의 기저로부터 확장된 기저에 대한 선형변환의 행렬표현 정리 W ≤ V W \le V W ≤ V n n n 차원 벡터공간 V V V γ = { v 1 , … , v k } \gamma = \left\{ \mathbf{v}_{1}, \dots ,\mathbf{v}_{k} \right\} γ = { v 1 , … , v k } W W W 순서기저 라고 하자. β = { v 1 , … , v k , v k + 1 , … v n } \beta = \left\{ \mathbf{v}_{1}, \dots, \mathbf{v}_{k}, \mathbf{v}_{k+1}, \dots \mathbf{v}_{n} \right\} β = { v 1 , … , v k , v k + 1 , … v n } γ \gamma γ 확장된 V V V 라고 하자. T : V → V T : V \to V T : V → V β \beta β T T T 행렬표현 은 다음과 같은 블록행렬 이다.
[ T ] β = [ A 1 A 2 O A 3 ]
\begin{bmatrix} T \end{bmatrix}_{\beta} = \begin{bmatrix} A_{1} & A_{2} \\ O & A_{3} \end{bmatrix}
[ T ] β = [ A 1 O A 2 A 3 ]
이때 A 1 = [ T ∣ W ] γ A_{1} = \begin{bmatrix} T|_{W} \end{bmatrix}_{\gamma} A 1 = [ T ∣ W ] γ T ∣ W : W → W T|_{W} : W \to W T ∣ W : W → W 축소사상 , A 2 A_{2} A 2 k × n − k k \times n-k k × n − k A 3 A_{3} A 3 n − k × n − k n-k \times n-k n − k × n − k O O O n − k × k n-k \times k n − k × k 영행렬 이다.
증명 T ∣ W : W → W T|_{W} : W \to W T ∣ W : W → W 축소사상 이라고 하자. [ T ] β = [ t i j ] \begin{bmatrix} T \end{bmatrix}_{\beta} = [t_{ij}] [ T ] β = [ t ij ]
[ T ] β = [ t 11 t 12 ⋯ t 1 n t 21 t 22 ⋯ t 2 n ⋮ ⋮ ⋱ ⋮ t n 1 t n 2 ⋯ t n n ]
\begin{align*}
\begin{bmatrix} T \end{bmatrix}_{\beta} =
\begin{bmatrix}
t_{11} & t_{12} & \cdots & t_{1n} \\
t_{21} & t_{22} & \cdots & t_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
t_{n1} & t_{n2} & \cdots & t_{nn}
\end{bmatrix}
\end{align*}
[ T ] β = t 11 t 21 ⋮ t n 1 t 12 t 22 ⋮ t n 2 ⋯ ⋯ ⋱ ⋯ t 1 n t 2 n ⋮ t nn
행렬표현을 찾기 위해서는 기저의 원소가 어떻게 매핑되는지를 알면 된다. 다시말해 [ T ] β \begin{bmatrix} T \end{bmatrix}_{\beta} [ T ] β t i 1 t_{i1} t i 1 T v 1 T\mathbf{v}_{1} T v 1 β \beta β v i \mathbf{v}_{i} v i T v 1 = T ∣ W v 1 T \mathbf{v}_{1} = T|_{W}\mathbf{v}_{1} T v 1 = T ∣ W v 1
∑ i = 1 n a i 1 v i = T v 1 = T ∣ W v 1 = ∑ i = 1 k a i 1 v i
\sum_{i=1}^{n} a_{i1} \mathbf{v}_{i} = T\mathbf{v}_{1} = T|_{W}\mathbf{v}_{1} = \sum_{i=1}^{k} a_{i1} \mathbf{v}_{i}
i = 1 ∑ n a i 1 v i = T v 1 = T ∣ W v 1 = i = 1 ∑ k a i 1 v i
따라서 a k + 1 = a k + 2 = ⋯ = a n = 0 a_{k+1} = a_{k+2} = \cdots = a_{n} = 0 a k + 1 = a k + 2 = ⋯ = a n = 0 1 ≤ i ≤ k 1\le i \le k 1 ≤ i ≤ k t i 1 = a i 1 t_{i1} = a_{i1} t i 1 = a i 1 k > i k \gt i k > i t i 1 = 0 t_{i1} = 0 t i 1 = 0
[ T ] β = [ a 11 t 12 ⋯ t 1 n a 21 t 22 ⋯ t 2 n ⋮ ⋮ ⋱ ⋮ a k 1 t k 2 ⋯ t k n 0 t k + 1 , 2 ⋯ t k + 1 , n ⋮ ⋮ ⋱ ⋮ 0 t m 2 ⋯ t m n ]
\begin{align*}
\begin{bmatrix} T \end{bmatrix}_{\beta} =
\begin{bmatrix}
a_{11} & t_{12} & \cdots & t_{1n} \\
a_{21} & t_{22} & \cdots & t_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{k1} & t_{k2} & \cdots & t_{kn} \\
0 & t_{k+1,2} & \cdots & t_{k+1,n} \\
\vdots & \vdots & \ddots & \vdots \\
0 & t_{m2} & \cdots & t_{mn}
\end{bmatrix}
\end{align*}
[ T ] β = a 11 a 21 ⋮ a k 1 0 ⋮ 0 t 12 t 22 ⋮ t k 2 t k + 1 , 2 ⋮ t m 2 ⋯ ⋯ ⋱ ⋯ ⋯ ⋱ ⋯ t 1 n t 2 n ⋮ t kn t k + 1 , n ⋮ t mn
같은 방법으로 k k k
[ T ] β = [ a 11 a 12 ⋯ a 1 k t 1 , k + 1 ⋯ t 1 n a 21 a 22 ⋯ a 2 k t 2 , k + 1 ⋯ t 2 n ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ a k 1 a k 2 ⋯ a k k t k , k + 1 ⋯ t k n 0 0 ⋯ 0 t k + 1 , k + 1 ⋯ t k + 1 , n ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ 0 t m , k + 1 ⋯ t m n ]
\begin{bmatrix} T \end{bmatrix}_{\beta} =
\begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1k} & t_{1,k+1} & \cdots & t_{1n} \\
a_{21} & a_{22} & \cdots & a_{2k} & t_{2,k+1} & \cdots & t_{2n} \\
\vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
a_{k1} & a_{k2} & \cdots & a_{kk} & t_{k,k+1} & \cdots & t_{kn} \\
0 & 0 & \cdots & 0 & t_{k+1,k+1} & \cdots & t_{k+1,n} \\
\vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & 0 & t_{m,k+1} & \cdots & t_{mn}
\end{bmatrix}
[ T ] β = a 11 a 21 ⋮ a k 1 0 ⋮ 0 a 12 a 22 ⋮ a k 2 0 ⋮ 0 ⋯ ⋯ ⋱ ⋯ ⋯ ⋱ ⋯ a 1 k a 2 k ⋮ a kk 0 ⋮ 0 t 1 , k + 1 t 2 , k + 1 ⋮ t k , k + 1 t k + 1 , k + 1 ⋮ t m , k + 1 ⋯ ⋯ ⋱ ⋯ ⋯ ⋱ ⋯ t 1 n t 2 n ⋮ t kn t k + 1 , n ⋮ t mn
이때 A 1 = [ a i j ] = [ T ∣ W ] γ A_{1} = [a_{ij}] = \begin{bmatrix} T|_{W} \end{bmatrix}_{\gamma} A 1 = [ a ij ] = [ T ∣ W ] γ A 2 = [ t 1 , k + 1 ⋯ t 1 n ⋮ ⋱ ⋮ t k , k + 1 ⋯ t k n ] A_{2} = \begin{bmatrix} t_{1,k+1} & \cdots & t_{1n} \\ \vdots & \ddots & \vdots \\ t_{k,k+1} & \cdots & t_{kn} \\ \end{bmatrix} A 2 = t 1 , k + 1 ⋮ t k , k + 1 ⋯ ⋱ ⋯ t 1 n ⋮ t kn A 3 = [ t k + 11 , k + 1 ⋯ t k + 1 , n ⋮ ⋱ ⋮ t m , k + 1 ⋯ t m n ] A_{3} = \begin{bmatrix} t_{k+11,k+1} & \cdots & t_{k+1,n} \\ \vdots & \ddots & \vdots \\ t_{m,k+1} & \cdots & t_{mn} \\ \end{bmatrix} A 3 = t k + 11 , k + 1 ⋮ t m , k + 1 ⋯ ⋱ ⋯ t k + 1 , n ⋮ t mn
[ T ] β = [ A 1 A 2 O A 3 ] = [ [ T ∣ W ] γ A 2 O A 3 ]
\begin{bmatrix} T \end{bmatrix}_{\beta} = \begin{bmatrix} A_{1} & A_{2} \\ O & A_{3} \end{bmatrix} = \begin{bmatrix} \begin{bmatrix} T|_{W} \end{bmatrix}_{\gamma} & A_{2} \\ O & A_{3} \end{bmatrix}
[ T ] β = [ A 1 O A 2 A 3 ] = [ [ T ∣ W ] γ O A 2 A 3 ]
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