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Study/pages/4.2 직선과 부분공간 위로의 사영.md index ad9b72e..962559b 100755 --- a/Major Study/pages/4.2 직선과 부분공간 위로의 사영.md +++ b/Major Study/pages/4.2 직선과 부분공간 위로의 사영.md @@ -9,6 +9,7 @@ deck:: Logseq/전공공부/선형대수학 - "가장 가깝다"와 "$e$ 가 직교"는 같은 말이다: 수직으로 내려간 발이 최단 거리라는 것, 이 절 전체가 이 한 문장 위에 서 있다. - 직관: 사영 $p$ 는 $b$ 의 "그 방향 성분"이다. - 📌 원문 (p.219): ((6a7a0415-779e-42a1-9b91-95340aa2264b)) + - 📌 원문 그림 4.3 (p.221): [🔗 PDF 위치](((6a7b0d35-505d-4e2b-89fb-270f4e5f86de))) - ![그림 4.3 직선과 부분공간 위로의 사영 (p.221)](../assets/4장_p221_그림4.3_직선_부분공간_사영.png) - (p.221, 그림 4.3) - 💡 **점 관점으로 보기** (끝점의 언어로 번역): 원점 $O$, $b$ 의 끝점을 $B$, $p$ 의 끝점을 $P$ 라 하자. @@ -91,6 +92,8 @@ deck:: Logseq/전공공부/선형대수학 - ② 그래서 내적 방정식이 정확히 $n$ 개 나온다: $a_1^{\mathrm{T}}(b - A\hat{x}) = 0,\ \ldots,\ a_n^{\mathrm{T}}(b - A\hat{x}) = 0$. - ③ 이 $n$ 개를 행렬로 묶는다: $a_i^{\mathrm{T}}$ 들을 행으로 쌓은 행렬이 바로 $A^{\mathrm{T}}$ 다. 그러면 $n$ 개의 식이 한 줄이 된다: - $$A^{\mathrm{T}}(b - A\hat{x}) = 0$$ + - 📌 원문 식 (4) (p.224): [🔗 PDF 위치](((6a7b0d0a-167b-4f1f-9889-09c8be19e7e0))) + - ![식 (4): n개의 직교 방정식이 정규방정식으로 (p.224)](../assets/4장_p224_식4_정규방정식_유도.png) - ④ 괄호를 풀어 이항하면 **정규방정식**(normal equation): - $$A^{\mathrm{T}}A\hat{x} = A^{\mathrm{T}}b$$ - 📌 원문 (p.225): ((6a7af250-44f8-4e4d-8c66-e774cd33390a)) @@ -123,6 +126,8 @@ deck:: Logseq/전공공부/선형대수학 - 🧩 $A^{\mathrm{T}}A$ **가 가역행렬일 필요충분조건은** $A$ **의 열이 선형독립인 것이다.** - 증명의 핵심은 4.1 연습문제 9 그대로: $\mathbf{N}(A^{\mathrm{T}}A) = \mathbf{N}(A)$. - $A^{\mathrm{T}}Ax = 0$ 이면 $x^{\mathrm{T}}$ 를 곱해 $\|Ax\|^2 = 0$ → $Ax = 0$ (식 (12), 지난 절에서 두 가지 방법으로 증명했던 그 사실). + - 📌 원문 식 (12) (p.227): [🔗 PDF 위치](((6a7b0d23-2b91-414c-a5e2-50a025e0a7ba))) + - ![식 (12): 한 줄 증명 (p.227)](../assets/4장_p227_식12_증명_한줄.png) - 열이 독립이면 $\mathbf{N}(A) = \{0\}$ → $\mathbf{N}(A^{\mathrm{T}}A) = \{0\}$ → 정사각행렬 $A^{\mathrm{T}}A$ 는 가역 (4.1 [6단계]의 "영공간이 0뿐인 정사각 = 가역"). - 열이 종속이면 같은 논리로 $A^{\mathrm{T}}A$ 도 특이행렬. 책의 2×2 예: $\begin{bmatrix} 1&2 \\ 1&2 \\ 0&0 \end{bmatrix}$ 는 $A^{\mathrm{T}}A = \begin{bmatrix} 2&4 \\ 4&8 \end{bmatrix}$ (특이), 3열을 $(0,0,1)$ 로 바꾸면 가역. - $A^{\mathrm{T}}A$ 의 프로필 정리: 크기 $n \times n$ 정사각, 대칭 ($(A^{\mathrm{T}}A)^{\mathrm{T}} = A^{\mathrm{T}}A$), 열 독립일 때 가역. "직사각행렬 문제는 항상 $A^{\mathrm{T}}A$ 로 이어진다"(p.227)는 경험칙을 기억할 것. diff --git a/Major Study/pages/hls__4장_직교성_1786280501491_0.md b/Major Study/pages/hls__4장_직교성_1786280501491_0.md index 5343671..79e58e5 100755 --- a/Major Study/pages/hls__4장_직교성_1786280501491_0.md +++ b/Major Study/pages/hls__4장_직교성_1786280501491_0.md @@ -246,4 +246,25 @@ file-path:: ../assets/4장_직교성_1786280501491_0.pdf hl-color:: yellow id:: 6a7af28d-1121-4a8f-94c5-3749f4744e14 hl-type:: area - hl-stamp:: 1786442380971 \ No newline at end of file + hl-stamp:: 1786442380971 +- [:span] + ls-type:: annotation + hl-page:: 18 + hl-color:: yellow + id:: 6a7b0d0a-167b-4f1f-9889-09c8be19e7e0 + hl-type:: area + hl-stamp:: 1786449160532 +- [:span] + ls-type:: annotation + hl-page:: 21 + hl-color:: yellow + id:: 6a7b0d23-2b91-414c-a5e2-50a025e0a7ba + hl-type:: area + hl-stamp:: 1786449187251 +- [:span] + ls-type:: annotation + hl-page:: 15 + hl-color:: yellow + id:: 6a7b0d35-505d-4e2b-89fb-270f4e5f86de + hl-type:: area + hl-stamp:: 1786449205469 \ No newline at end of file