From ce806ff1c1da6d6f5b6c7f2b85ae5b2a4c178958 Mon Sep 17 00:00:00 2001 From: songyc macbook Date: Mon, 10 Aug 2026 01:37:55 +0900 Subject: [PATCH] =?UTF-8?q?4.1=20=EC=97=B0=EC=8A=B5=EB=AC=B8=EC=A0=9C=20?= =?UTF-8?q?=EC=A0=84=EB=A9=B4=20=EB=B3=B4=EA=B0=95:=20=EA=B0=9C=EB=85=90?= =?UTF-8?q?=20=EC=83=81=EA=B8=B0=C2=B7=EB=8B=A8=EA=B3=84=EB=B3=84=20?= =?UTF-8?q?=ED=92=80=EC=9D=B4=C2=B7=EA=B2=80=EC=82=B0=20=EA=B5=AC=EC=A1=B0?= =?UTF-8?q?=EB=A1=9C=20=EC=9E=AC=EC=9E=91=EC=84=B1=20+=20=EC=9E=90?= =?UTF-8?q?=EC=9E=91=20=EA=B7=B8=EB=A6=BC=202=EA=B0=9C?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit - 문제 2·3·6·9·12·17 전부 '🔁 개념 상기 → 단계별 → 검산 → 핵심 문장' 구조로 확장 - 문제 3: 직교 강제 쌍 판정법 + (b) 함정 주의 + (d) 랭크1 구성법 + (e) 별해 - 문제 6: y 계산 연립 전 과정 + 프레드홀름 대안 직관 - 문제 9: 대안 증명(‖Ax‖²) + N(AᵀA)=N(A) 양방향 + 4.2~4.3 예고 - 자작 SVG→PNG 2개: 문제 2 (랭크 2 그림 4.1 변형), 문제 12 (x 분해 그림) Co-Authored-By: Claude Fable 5 Claude-Session: https://claude.ai/code/session_01Ufr6wUhJAzBaqcaBiRJ1xk 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모든 x가 행공간 소속 + (xₙ = 0) + + + 출력 공간 ℝ³ (m = 3) + + 열공간 C(A) + 차원 2 (평면) + + + 좌영공간 N(Aᵀ) + (차원 1, 평면과 ⊥) + + + 차원 검산: 왼쪽 2 + 0 = 2 = n ✓, 오른쪽 2 + 1 = 3 = m ✓ + diff --git a/Major Study/pages/4.1 벡터와 부분공간의 직교성.md b/Major Study/pages/4.1 벡터와 부분공간의 직교성.md index 6c3eed1..acf6957 100755 --- a/Major Study/pages/4.1 벡터와 부분공간의 직교성.md +++ b/Major Study/pages/4.1 벡터와 부분공간의 직교성.md @@ -151,55 +151,132 @@ deck:: Logseq/전공공부/선형대수학 - 행공간 기저 $r$ 개 + 영공간 기저 $n-r$ 개 = $n$ 개가 $\mathbb{R}^n$ 의 기저이므로, 어떤 $x$ 든 두 성분의 합으로 (유일하게) 쪼개진다. - ### ✏️ 연습문제 풀이 (솔루션 참고) - **문제 2**: 랭크 $r = 2$ 인 $3 \times 2$ 행렬 기준으로 [그림 4.1]을 다시 작성하라. 어떤 부분공간이 $\mathbf{Z}$ (영벡터만 포함)가 되는가? $\mathbb{R}^2$ 의 임의의 벡터 $x$ 의 영공간 부분은 $x_n = ?$ - - ![그림 4.1 직교여공간과 x의 분해 (p.212)](../assets/4장_p212_그림4.1_직교여공간_그림.png) - - $n = 2$, $r = 2$: 열 2개가 독립이므로 **영공간** $= \mathbf{Z} = \{0\}$. 따라서 모든 $x$ 에 대해 $x_n = 0$. - - 행공간 $= \mathbb{R}^2$ 전체 ($r = n$). 그림 왼쪽($\mathbb{R}^2$)에서 영공간 상자가 점 하나로 쪼그라든다. - - 오른쪽($\mathbb{R}^3$): 열공간은 2차원 평면, 좌영공간은 그 평면과 직교하는 직선 ($m - r = 1$). - - 핵심 문장: 랭크가 꽉 차는 쪽의 영공간은 $\{0\}$ 으로 사라진다. + - 🔁 개념 상기: + - [그림 4.1]은 4공간의 지도였다: 왼쪽 $\mathbb{R}^n$ 에 행공간 ⊥ 영공간, 오른쪽 $\mathbb{R}^m$ 에 열공간 ⊥ 좌영공간 ([4단계]). + - 4공간의 차원은 $r,\ n-r,\ r,\ m-r$ (3장 기본정리 1). 그림을 다시 그리는 문제는 결국 이 네 숫자를 채우는 문제다. + - ![그림 4.1 직교여공간과 x의 분해 (p.212)](../assets/4장_p212_그림4.1_직교여공간_그림.png) + - 1단계, 네 차원 채우기: $m = 3$, $n = 2$, $r = 2$ 를 공식에 넣는다. + - 행공간 $r = 2$ / 영공간 $n - r = 2 - 2 = 0$ / 열공간 $r = 2$ / 좌영공간 $m - r = 3 - 2 = 1$. + - 2단계, 차원 0의 정체: 차원이 0인 부분공간은 영벡터 하나만 담은 공간이다. 답: **영공간이** $\mathbf{Z} = \{0\}$ 이 된다. + - 직관으로도 확인: 열이 2개뿐인데 랭크가 2 = 두 열이 선형독립 = $Ax = 0$ 의 해가 $x = 0$ 뿐 (독립의 정의 그대로). + - 3단계, 그림 다시 그리기: 왼쪽 $\mathbb{R}^2$ 는 행공간이 평면 전체를 차지하고, 영공간은 원점 한 점으로 수축한다. 오른쪽 $\mathbb{R}^3$ 는 열공간 평면(2차원)과 그에 수직인 좌영공간 직선(1차원). + - ![문제 2: 랭크 2인 3×2 행렬의 그림 4.1](../assets/4장_4.1_문제2_그림4.1_랭크2_3x2.png) + - 4단계, $x_n$ 답하기: 분해 $x = x_r + x_n$ ([5단계])에서 $x_n$ 은 영공간에서 골라야 하는데, 영공간에 $0$ 밖에 없다. + - 따라서 모든 $x$ 에 대해 $x_n = 0$, 즉 $x = x_r$. $\mathbb{R}^2$ 의 모든 벡터가 통째로 행공간 소속이다. + - 핵심 문장: 랭크가 열 개수와 같으면 ($r = n$, 최대 열 랭크) 영공간이 $\{0\}$ 으로 사라지고, 모든 $x$ 의 영공간 성분은 $x_n = 0$ 이다. - **문제 3** (이 절의 대표 문제): 다음 성질을 만족하는 행렬을 구성하라. 불가능하면 이유를 설명하라. - 📌 문제 지문 (p.215): - ![4.1 연습문제 3 지문 (p.215)](../assets/4장_p215_4.1_연습문제3_지문.png) - - (a) 열공간이 $(1,2,-3), (2,-3,5)$ 를 포함, 영공간이 $(1,1,1)$ 을 포함: **가능**. - - 열공간 조건은 $\mathbb{R}^m$ (출력) 얘기, 영공간 조건은 $\mathbb{R}^n$ (입력) 얘기라 서로 직교 제약이 없다. - - 구성: 주어진 두 벡터를 1·2열로 쓰고, $(1,1,1) \in \mathbf{N}(A)$ 가 되도록 3열을 $-(\text{1열}) - (\text{2열})$ 로 잡는다. + - 🔁 개념 상기: 이 문제의 판정 도구는 두 개뿐이다. + - **직교가 강제되는 쌍**: 행공간 ⊥ 영공간 (둘 다 $\mathbb{R}^n$ 소속), 열공간 ⊥ 좌영공간 (둘 다 $\mathbb{R}^m$ 소속). 이 쌍에 들어갈 두 벡터의 내적이 0이 아니면 그런 행렬은 **존재 자체가 불가능**하다. + - **직교 제약이 없는 조합**: 열공간과 영공간처럼 서로 다른 공간($\mathbb{R}^m$ 과 $\mathbb{R}^n$)에 사는 조합은 아무 제약이 없어 자유롭게 구성할 수 있다 ([3단계] ⚠️의 "직교는 같은 공간에 사는 것끼리"가 바로 이 판정의 근거). + - 풀이 순서: ① 조건에 등장하는 벡터가 각각 어느 부분공간 소속인지 번역 → ② 직교 강제 쌍인지 확인 → ③ 강제면 내적 검사, 아니면 직접 구성. + - (a) 열공간이 $(1,2,-3), (2,-3,5)$ 를 포함하고 영공간이 $(1,1,1)$ 을 포함: **가능**. + - ① 번역: 열공간 조건은 출력 쪽($\mathbb{R}^3$), 영공간 조건은 입력 쪽($\mathbb{R}^3$이지만 다른 세계) 얘기다. + - ② 판정: 열공간 vs 영공간은 직교 강제 쌍이 아니다 → 구성 가능성 열림. + - ③ 구성 전략: "열공간이 어떤 벡터를 포함"은 그 벡터를 그냥 열로 넣으면 해결된다. "영공간이 $(1,1,1)$ 을 포함"은 $A(1,1,1)^{\mathrm{T}} = 0$, 즉 **1열 + 2열 + 3열 = 0** 이라는 뜻 ($Ax$ = 열들의 결합, 1장). + - 그래서 3열을 $-(\text{1열}) - (\text{2열})$ 로 잡으면 끝: - $$A = \begin{bmatrix} 1&2&-3 \\ 2&-3&1 \\ -3&5&-2 \end{bmatrix}$$ - - 검산: 1열 + 2열 + 3열 $= 0$ 이므로 $A(1,1,1)^{\mathrm{T}} = 0$ ✓. - - (b) **행공간**이 그 두 벡터를 포함, 영공간이 $(1,1,1)$ 을 포함: **불가능**. - - 행공간 ⊥ 영공간이어야 하는데 $(2,-3,5) \cdot (1,1,1) = 4 \neq 0$. + - 검산: 각 행에서 $1+2-3 = 0$, $2-3+1 = 0$, $-3+5-2 = 0$ ✓ ($A(1,1,1)^{\mathrm{T}} = 0$). + - (b) **행공간**이 그 두 벡터를 포함하고 영공간이 $(1,1,1)$ 을 포함: **불가능**. + - ① 번역: 이번엔 행공간과 영공간, 둘 다 $\mathbb{R}^3$ (입력 세계) 소속. + - ② 판정: 직교 강제 쌍이다! 행공간에 들어갈 벡터와 영공간에 들어갈 벡터는 반드시 내적 0. + - ③ 내적 검사: $(1,2,-3) \cdot (1,1,1) = 1 + 2 - 3 = 0$ ✓ 통과. 그런데 $(2,-3,5) \cdot (1,1,1) = 2 - 3 + 5 = 4 \neq 0$ ✗. + - ⚠️ 함정 주의: 첫 벡터는 통과한다! 하나만 검사하고 "가능"이라 답하기 쉽다. 직교는 **모든** 쌍에 대해 성립해야 하므로 전부 검사해야 한다. - (c) $Ax = (1,1,1)$ 의 해가 존재하고 $A^{\mathrm{T}}(1,0,0)^{\mathrm{T}} = 0$: **불가능**. - - 해가 존재한다 = $(1,1,1) \in \mathbf{C}(A)$. 그리고 $(1,0,0) \in \mathbf{N}(A^{\mathrm{T}})$. - - 열공간 ⊥ 좌영공간인데 $(1,1,1) \cdot (1,0,0) = 1 \neq 0$. - - (d) 모든 행이 모든 열과 직교하는 0 아닌 행렬: **가능**. - - $$A = \begin{bmatrix} 1&-1 \\ 1&-1 \end{bmatrix}$$ - - 행 $(1,-1)$ 과 열 $(1,1), (-1,-1)$ 의 내적이 전부 0. - - 💡 이 조건은 $A \cdot A = 0$ (각 성분이 행·열 내적)과 같은 말이다. 실제로 $A^2 = 0$ ✓. + - ① 번역: "$Ax = b$ 의 해 존재" ⟺ $b \in \mathbf{C}(A)$ (3.3절: 해 존재 조건). 그러니 $(1,1,1) \in \mathbf{C}(A)$. "$A^{\mathrm{T}}y = 0$" ⟺ $y \in \mathbf{N}(A^{\mathrm{T}})$. 그러니 $(1,0,0) \in \mathbf{N}(A^{\mathrm{T}})$. + - ② 판정: 열공간 vs 좌영공간, 직교 강제 쌍 (둘 다 $\mathbb{R}^3$ 출력 세계). + - ③ 내적 검사: $(1,1,1) \cdot (1,0,0) = 1 \neq 0$ ✗. 존재할 수 없다. + - 💡 문장을 부분공간 언어로 번역하는 훈련이 이 문제의 진짜 목적이다. "해가 존재한다"를 보자마자 "$b$ 가 열공간에 있다"로 읽을 수 있어야 한다. + - (d) 모든 행이 모든 열과 직교하는 영행렬 아닌 행렬: **가능**. + - ① 번역: 행공간 ⊥ 열공간이 되어야 한다. 이 둘은 직교 강제 쌍이 **아니고**, 애초에 같은 공간에 살려면 $m = n$ (정사각)이어야 한다. + - ② 구성 전략: 랭크 1 행렬 $A = uv^{\mathrm{T}}$ 는 열공간 = $u$ 방향, 행공간 = $v$ 방향이다 (3장). $u \perp v$ 로 고르면 끝: $u = (1,1)$, $v = (1,-1)$. + - $$A = \begin{bmatrix} 1 \\ 1 \end{bmatrix}\begin{bmatrix} 1&-1 \end{bmatrix} = \begin{bmatrix} 1&-1 \\ 1&-1 \end{bmatrix}$$ + - 검산: 행 $(1,-1)$ 과 1열 $(1,1)$ 의 내적 $1-1 = 0$ ✓, 2열 $(-1,-1)$ 과도 $-1+1 = 0$ ✓. + - 💡 재미있는 동치: $(A^2)_{ij} = (i\text{행}) \cdot (j\text{열})$ 이므로 "모든 행 ⊥ 모든 열" ⟺ $A^2 = 0$. 위 행렬로 확인하면 정말 $A^2 = 0$ ✓. - (e) 모든 열의 합 = 영벡터, 모든 행의 합 = 성분이 전부 1인 행: **불가능**. - - 열들의 합이 0 ⟺ $A \cdot (1, \ldots, 1)^{\mathrm{T}} = 0$ ⟺ 모든 성분이 1인 벡터가 **영공간**에 있다. - - 행들의 합이 $(1, \ldots, 1)$ ⟺ 모든 성분이 1인 벡터가 행들의 결합, 즉 **행공간**에 있다. - - 영공간과 행공간에 동시에 속하는 0 아닌 벡터는 없다. 모순. - - 더 쉬운 검산: 전체 성분 총합을 열 방향으로 세면 0, 행 방향으로 세면 $n$. 같아야 하는데 다르다. - - 핵심 문장: 구성 가능/불가능의 판정 기준은 전부 "직교해야 할 쌍(행공간·영공간, 열공간·좌영공간)이 직교를 깨는가"다. + - ① 번역 (열 조건): 열들을 전부 더한 것은 $A$ 에 $(1, \ldots, 1)^{\mathrm{T}}$ 를 곱한 것. 이것이 0 ⟺ 모든 성분이 1인 벡터가 **영공간**에 있다. + - ① 번역 (행 조건): 행들을 전부 더한 것은 계수를 전부 1로 준 행들의 선형결합. 이것이 $(1, \ldots, 1)$ ⟺ 모든 성분이 1인 벡터가 **행공간**에 있다. + - ② 판정: 같은 벡터 $(1, \ldots, 1)$ 이 영공간과 행공간에 동시에 들어가야 한다. 직교하는 두 공간의 공통 벡터는 영벡터뿐인데 ([2단계]) 이 벡터는 0이 아니다. 모순. + - 별해 (총합 세기): 행렬의 모든 성분의 총합을 두 방식으로 센다. 열 방향으로 모으면 (열 합이 0이니) 총합 0, 행 방향으로 모으면 (행 합이 성분 1짜리 행이니) 총합 $n$. $0 \neq n$ 이므로 모순. + - 핵심 문장: 구성 가능/불가능 판정 = "그 조건이 직교 강제 쌍(행공간·영공간 / 열공간·좌영공간)에 내적 0 아닌 두 벡터를 넣으려 하는가" 검사. - **문제 6**: 세 방정식 $x + 2y + 2z = 5$, $2x + 2y + 3z = 5$, $3x + 4y + 5z = 9$ 에 $y_1, y_2, y_3$ 을 곱해 더했을 때 $0 = 1$ 이 되게 하는 $y$ 를 구하라. $y$ 는 어떤 부분공간에 있는가? - - $y = (1, 1, -1)$: 1행 + 2행 − 3행을 계산하면 좌변의 $x, y, z$ 계수가 전부 0이 되고 우변은 $5 + 5 - 9 = 1$. - - 즉 $0 = 1$, 모순이므로 이 연립방정식은 해가 없다. - - $y$ 는 계수행렬 $A$ 의 행들을 결합해 영행을 만드는 계수이므로 **좌영공간** $\mathbf{N}(A^{\mathrm{T}})$ 의 벡터다. - - 부분공간 언어로: 해가 있으려면 모든 $y \in \mathbf{N}(A^{\mathrm{T}})$ 에 대해 $y^{\mathrm{T}}b = 0$ 이어야 하는데 (열공간 ⊥ 좌영공간), 여기선 $y^{\mathrm{T}}b = 1$. - - 💡 [문제 7]이 이를 일반화한 것이 **프레드홀름 대안**(Fredholm's alternative): 다음 둘 중 정확히 하나만 해를 갖는다. ① $Ax = b$, ② $A^{\mathrm{T}}y = 0$ 이면서 $y^{\mathrm{T}}b = 1$. - - 핵심 문장: "해 없음"의 증인이 바로 좌영공간 벡터 $y$ 다 ($y^{\mathrm{T}}b \neq 0$ 인 $y$). - - **문제 9**: $A^{\mathrm{T}}Ax = 0$ 이면 $Ax = 0$ 인 이유는? - - $Ax$ 는 항상 $A$ 의 **열공간**에 있다. - - $A^{\mathrm{T}}(Ax) = 0$ 이면 $Ax$ 는 $A^{\mathrm{T}}$ 의 영공간, 즉 **좌영공간**에도 있다. - - 두 공간은 직교하므로 $Ax$ 는 자기 자신과 직교한다: $\|Ax\|^2 = 0$, 곧 $Ax = 0$. - - 결론: $A^{\mathrm{T}}A$ 와 $A$ 는 **동일한 영공간**을 갖는다 ($Ax=0$ 이면 $A^{\mathrm{T}}Ax=0$ 은 자명하고, 역방향이 방금 증명한 것). - - 💡 수위 조절: 이 사실은 4.2~4.3절에서 $A^{\mathrm{T}}A$ 가 가역임을 보일 때 핵심 열쇠로 재등장한다. 지금은 결론 문장만 기억하면 된다. - - **문제 12**: $A = \begin{bmatrix} 1&-1 \\ 0&0 \end{bmatrix}$, $x = (2, 0)$ 에 대해 $x_r, x_n$ 을 찾아라 ([그림 4.1] 형태, 문제 2의 그림 참고). - - 행공간: $(1, -1)$ 방향 직선. 영공간: $x_1 = x_2$, 즉 $(1, 1)$ 방향 직선. 서로 직교 ✓. - - $x = a(1,-1) + b(1,1)$ 로 놓으면: $a + b = 2$, $-a + b = 0$ → $a = b = 1$. - - $$x_r = (1, -1), \qquad x_n = (1, 1)$$ - - 검산: $Ax = (2, 0)$, $Ax_r = (1+1, 0) = (2, 0)$, 정말 $Ax = Ax_r$ ✓. - - 핵심 문장: 분해는 "행공간 기저와 영공간 기저로 $x$ 를 표현하는 연립방정식"으로 구한다 (일반 도구는 4.2의 사영). + - 🔁 개념 상기: + - 좌영공간 $\mathbf{N}(A^{\mathrm{T}})$ 의 원소 = "$A$ 의 행들을 결합해 영행을 만드는 계수 묶음" (3.5절, 그리고 3.5 연습문제 3에서 직접 구해봤던 그것). + - 해 존재 조건의 직교 버전 (이 절의 새 관점): $Ax = b$ 가 풀린다 ⟺ $b \in \mathbf{C}(A)$ ⟺ $b$ 가 **모든 좌영공간 벡터와 직교** (열공간 ⊥ 좌영공간이고, 좌영공간이 열공간의 직교여공간이라 "좌영공간과 다 직교하면 열공간 안"이 성립). + - 1단계, 문제 번역: 방정식들에 $y_1, y_2, y_3$ 을 곱해 더하는 것 = 행들의 선형결합. "좌변이 0, 우변이 1"이 되려면: + - 좌변 조건: $y_1(1,2,2) + y_2(2,2,3) + y_3(3,4,5) = (0,0,0)$, 즉 $y \in \mathbf{N}(A^{\mathrm{T}})$. + - 우변 조건: $5y_1 + 5y_2 + 9y_3 = 1$, 즉 $y^{\mathrm{T}}b = 1$. + - 2단계, $y$ 계산 (성분별 연립): + - $x$ 계수: $y_1 + 2y_2 + 3y_3 = 0$ + - $y$ 계수: $2y_1 + 2y_2 + 4y_3 = 0$ + - 둘째에서 첫째를 빼면: $y_1 + y_3 = 0$ → $y_1 = -y_3$. + - 첫째에 대입: $-y_3 + 2y_2 + 3y_3 = 0$ → $y_2 = -y_3$. + - $z$ 계수 검산: $2y_1 + 3y_2 + 5y_3 = -2y_3 - 3y_3 + 5y_3 = 0$ ✓ (자동 만족, 좌영공간이 1차원 직선이라는 뜻). + - $y_3 = -1$ 로 고르면 $y = (1, 1, -1)$. + - 3단계, 확인: 1행 + 2행 − 3행: + - 좌변: $(1+2-3,\ 2+2-4,\ 2+3-5) = (0, 0, 0)$ ✓ + - 우변: $5 + 5 - 9 = 1$ ✓ → 정말 $0 = 1$. 모순이므로 **해가 없다**. + - 4단계, 부분공간 언어로 결론: $y = (1,1,-1)$ 은 **좌영공간** $\mathbf{N}(A^{\mathrm{T}})$ 의 벡터다. + - 해가 있으려면 $b$ 는 열공간에 있어야 하고, 그러면 모든 좌영공간 벡터와 직교해야 한다 ($y^{\mathrm{T}}b = 0$). + - 그런데 $y^{\mathrm{T}}b = 1 \neq 0$. $b = (5,5,9)$ 가 열공간 밖에 있다는 움직일 수 없는 증거다. + - 💡 [문제 7]이 이를 일반화한 것이 **프레드홀름 대안**(Fredholm's alternative): 다음 둘 중 **정확히 하나**만 해를 갖는다. ① $Ax = b$, ② $A^{\mathrm{T}}y = 0$ 이면서 $y^{\mathrm{T}}b = 1$. + - 직관: $b$ 는 열공간 안에 있거나(①이 성립), 밖에 있다. 밖에 있으면 $b$ 의 좌영공간 방향 성분이 남아 있어서 그것을 증언하는 $y$ (②)가 반드시 존재한다. 둘 다 성립할 수는 없다 (①이면 $y^{\mathrm{T}}b = y^{\mathrm{T}}Ax = (A^{\mathrm{T}}y)^{\mathrm{T}}x = 0 \neq 1$). + - 핵심 문장: "해 없음"의 증인은 좌영공간 벡터다. $A^{\mathrm{T}}y = 0$ 인데 $y^{\mathrm{T}}b \neq 0$ 인 $y$ 하나가 해 없음을 증명한다. + - **문제 9**: $A^{\mathrm{T}}Ax = 0$ 이면 $Ax = 0$ 인 이유는? ($A^{\mathrm{T}}A$ 와 $A$ 의 영공간 비교) + - 🔁 개념 상기: + - $Ax$ 의 정체: 열들의 선형결합. 그래서 어떤 $x$ 를 넣든 $Ax$ 는 항상 **열공간** $\mathbf{C}(A)$ 안에 있다 (1장부터의 기본기). + - $\mathbf{N}(A^{\mathrm{T}})$ 의 정체: $A^{\mathrm{T}}(\cdot) = 0$ 을 만족하는 벡터들, 즉 좌영공간. + - 직교하는 두 부분공간의 공통 벡터는 영벡터뿐 ([2단계]). 이 세 가지를 조립하면 끝나는 문제다. + - 1단계: $Ax$ 는 (누가 뭐래도) 열공간 $\mathbf{C}(A)$ 안에 있다. + - 2단계: 가정 $A^{\mathrm{T}}Ax = 0$ 을 괄호만 바꿔 읽는다: $A^{\mathrm{T}}(Ax) = 0$. 벡터 $Ax$ 가 $A^{\mathrm{T}}$ 에게 죽는다 = $Ax$ 는 **좌영공간** $\mathbf{N}(A^{\mathrm{T}})$ 안에 있다. + - 3단계: 열공간 ⊥ 좌영공간인데 $Ax$ 가 양쪽에 다 있다 → 공통 벡터는 영벡터뿐 → $Ax = 0$. + - 대안 증명 (한 줄 계산, 나중에 4.3에서도 쓰는 트릭): 가정의 양변에 $x^{\mathrm{T}}$ 를 곱하면 + - $$x^{\mathrm{T}}A^{\mathrm{T}}Ax = (Ax)^{\mathrm{T}}(Ax) = \|Ax\|^2 = 0$$ + - 크기가 0인 벡터는 영벡터뿐이므로 $Ax = 0$. ("자기 자신과 직교하는 벡터는 0"의 계산 버전.) + - 결론: $\mathbf{N}(A^{\mathrm{T}}A) = \mathbf{N}(A)$. + - 방향 ⊇: $Ax = 0$ 이면 양변에 $A^{\mathrm{T}}$ 를 곱해 $A^{\mathrm{T}}Ax = 0$ (자명). + - 방향 ⊆: 방금 증명한 것. + - 두 행렬은 영공간이 같으니 랭크도 같다: $\mathrm{rank}(A^{\mathrm{T}}A) = \mathrm{rank}(A)$ (둘 다 열 개수 $n$ 에서 영공간 차원을 뺀 것). + - 💡 수위 조절: 이 결론은 4.2~4.3절의 심장이다. $A$ 의 열이 독립이면 ($\mathbf{N}(A) = \{0\}$) 정사각행렬 $A^{\mathrm{T}}A$ 도 영공간이 $\{0\}$ → **가역**. 그래서 최소제곱 방정식 $A^{\mathrm{T}}A\hat{x} = A^{\mathrm{T}}b$ 를 풀 수 있게 된다. 지금은 "$A^{\mathrm{T}}A$ 와 $A$ 는 영공간이 같다"까지만 기억하면 충분하다. + - 핵심 문장: $Ax$ 가 열공간과 좌영공간에 동시에 속하면 $Ax = 0$. 그래서 $\mathbf{N}(A^{\mathrm{T}}A) = \mathbf{N}(A)$. + - **문제 12**: $A = \begin{bmatrix} 1&-1 \\ 0&0 \end{bmatrix}$, $x = (2, 0)$ 에 대해 적절한 $x_r, x_n$ 을 찾아 [그림 4.1]을 그려라. + - 🔁 개념 상기: + - [5단계]의 분해 $x = x_r + x_n$ 을 손으로 직접 해보는 문제다. + - 분해가 가능한 근거는 [6단계]: 행공간 기저 + 영공간 기저 = $\mathbb{R}^n$ 의 기저. 그러니 "$x$ 를 그 기저로 표현하는 연립방정식"을 풀면 분해가 나온다. + - 1단계, 두 공간부터 구하기: + - 행공간: 0 아닌 행이 $(1, -1)$ 하나뿐 → $(1, -1)$ 방향 직선 (랭크 1). + - 영공간: $x_1 - x_2 = 0$, 즉 $x_1 = x_2$ → $(1, 1)$ 방향 직선. + - 직교 확인: $(1,-1) \cdot (1,1) = 1 - 1 = 0$ ✓ (기본정리 2가 약속한 대로). + - 2단계, $x$ 를 두 기저로 표현: $x = a(1,-1) + b(1,1)$ 로 놓고 성분별로 쓴다. + - 1성분: $a + b = 2$ + - 2성분: $-a + b = 0$ → $b = a$ + - 대입: $2a = 2$ → $a = 1$, $b = 1$. + - 3단계, 답: + - $$x_r = (1, -1), \qquad x_n = (1, 1)$$ + - ![문제 12: x = (2,0)의 분해 그림](../assets/4장_4.1_문제12_x분해.png) + - 4단계, 검산 3종 세트: + - 합: $x_r + x_n = (1,-1) + (1,1) = (2, 0) = x$ ✓ + - 직교: $x_r \cdot x_n = 1 - 1 = 0$ ✓ + - $A$ 가 보는 것: $Ax = (2-0,\ 0) = (2,0)$ 이고 $Ax_r = (1+1,\ 0) = (2,0)$. 정말 $Ax = Ax_r$ ✓ ($x_n$ 은 $A$ 에게 투명인간). + - 보너스 (그림 4.1 캡션의 식): $\|x\|^2 = \|x_r\|^2 + \|x_n\|^2$ 확인: $4 = 2 + 2$ ✓. 직교 분해라서 피타고라스가 성립한다 ([1단계]와 연결). + - 핵심 문장: 분해 계산법 = "행공간 기저와 영공간 기저의 결합으로 $x$ 를 표현하는 연립방정식 풀기". (계수를 자동으로 주는 일반 도구가 4.2의 사영이다.) - **문제 17**: $\mathbb{R}^3$ 에서 $\mathbf{S}^{\perp}$ 구하기 3연발. - - $\mathbf{S} = \{0\}$ 이면: $\mathbf{S}^{\perp} = \mathbb{R}^3$ 전체 (영벡터와는 모든 벡터가 직교). - - $\mathbf{S} = (1,1,1)$ 로 생성되는 직선이면: $\mathbf{S}^{\perp}$ 는 평면 $x + y + z = 0$. 기저는 $(1,-1,0), (1,0,-1)$. - - $\mathbf{S} = (1,1,1), (1,1,-1)$ 로 생성되면: 두 조건 $x+y+z=0$, $x+y-z=0$ 을 빼면 $z = 0$, 그러면 $x = -y$ → $\mathbf{S}^{\perp}$ 는 $(1, -1, 0)$ 방향 직선. - - 핵심 문장: $\mathbf{S}^{\perp}$ 구하기 = 생성 벡터들을 **행으로 넣은 행렬의 영공간** 구하기. 차원 검산: $\dim \mathbf{S} + \dim \mathbf{S}^{\perp} = 3$ 이 세 경우 모두 성립 ($0+3, 1+2, 2+1$). \ No newline at end of file + - 🔁 개념 상기: + - $\mathbf{S}^{\perp}$ 의 정의: $\mathbf{S}$ 의 **모든** 벡터와 직교하는 벡터 전부의 집합 ([4단계]). + - 실전 계산법은 기본정리 2를 거꾸로 쓰는 것: 생성 벡터들을 **행으로** 넣어 행렬 $A$ 를 만들면, "$\mathbf{S}$ 의 모든 벡터와 직교" = "$A$ 의 모든 행과 직교" = **영공간** $\mathbf{N}(A)$. 즉 $\mathbf{S}^{\perp} = \mathbf{N}(A)$. + - 검산 무기: $\dim \mathbf{S} + \dim \mathbf{S}^{\perp} = n$ (직교여공간은 차원을 남김없이 나눠 가진다). + - 케이스 1: $\mathbf{S} = \{0\}$. + - "$0$ 과 직교"는 모든 벡터가 만족한다 ($v^{\mathrm{T}}0 = 0$ 은 항상 참). + - $\mathbf{S}^{\perp} = \mathbb{R}^3$ 전체. 차원 검산: $0 + 3 = 3$ ✓. + - 케이스 2: $\mathbf{S} = (1,1,1)$ 로 생성되는 직선. + - $A = \begin{bmatrix} 1&1&1 \end{bmatrix}$ 의 영공간: 조건은 방정식 하나 $x + y + z = 0$ → 평면. + - 기저 (자유변수 $y, z$ 에 1 넣기): $(-1, 1, 0)$ 과 $(-1, 0, 1)$. (부호를 바꾼 $(1,-1,0), (1,0,-1)$ 도 물론 기저.) + - 차원 검산: $1 + 2 = 3$ ✓. 그림으로는 직선 $(1,1,1)$ 에 수직인 평면. + - 케이스 3: $\mathbf{S} = (1,1,1), (1,1,-1)$ 로 생성되는 평면. + - $A = \begin{bmatrix} 1&1&1 \\ 1&1&-1 \end{bmatrix}$ 의 영공간을 소거로 구한다: 2행 − 1행 → $(0, 0, -2)$, 즉 $z = 0$. + - 1행에 대입: $x + y = 0$. 피벗변수 $x, z$ / 자유변수 $y$ → 특별해 $(-1, 1, 0)$. + - $\mathbf{S}^{\perp}$ 는 $(1, -1, 0)$ 방향 직선. 차원 검산: $2 + 1 = 3$ ✓. + - 직교 확인: $(1,-1,0) \cdot (1,1,1) = 0$ ✓, $(1,-1,0) \cdot (1,1,-1) = 0$ ✓. + - 💡 세 케이스를 나란히 보면 규칙이 보인다: $\mathbf{S}$ 가 커질수록 ($0 \to 1 \to 2$차원) $\mathbf{S}^{\perp}$ 는 정확히 그만큼 작아진다 ($3 \to 2 \to 1$차원). 직교여공간은 시소 관계다. + - 핵심 문장: $\mathbf{S}^{\perp}$ 구하기 = 생성 벡터들을 행으로 넣은 행렬의 **영공간** 구하기. 검산은 $\dim \mathbf{S} + \dim \mathbf{S}^{\perp} = n$. \ No newline at end of file