From ce73a26a36b78dc86fb8473b4048860f531f475d Mon Sep 17 00:00:00 2001 From: songyc macbook Date: Thu, 6 Aug 2026 20:19:47 +0900 Subject: [PATCH] =?UTF-8?q?SVG=20=EC=9E=84=EB=B2=A0=EB=93=9C=20=EB=A0=8C?= =?UTF-8?q?=EB=8D=94=EB=A7=81=20=EB=AC=B8=EC=A0=9C=20=EC=88=98=EC=A0=95(PN?= =?UTF-8?q?G=20=EB=B3=80=ED=99=98=EB=B3=B8=EC=9C=BC=EB=A1=9C=20=EA=B5=90?= =?UTF-8?q?=EC=B2=B4)=20+=203=EC=9E=A5=20=EC=B9=B4=EB=93=9C=ED=99=94=20?= =?UTF-8?q?=EB=B0=98=EC=98=81=20+=20CLAUDE.md=20=EB=B8=94=EB=A1=9D=20?= =?UTF-8?q?=EB=B6=84=EB=A6=AC=C2=B7PNG=20=EC=9E=84=EB=B2=A0=EB=93=9C=20?= =?UTF-8?q?=EA=B7=9C=EC=B9=99?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-Authored-By: Claude Fable 5 Claude-Session: https://claude.ai/code/session_01G8rNxLqi5rYjRamC8uwiJW --- CLAUDE.md | 2 ++ Major Study/assets/2장_2.5_격자_경계조건.png | Bin 0 -> 51007 bytes Major Study/assets/2장_2.5_예제_해_비교.png | Bin 0 -> 83987 bytes Major Study/assets/2장_2.5_이차차분_직관.png | Bin 0 -> 96492 bytes Major Study/assets/2장_2.5_차분_세가지.png | Bin 0 -> 100898 bytes .../assets/3장_3.1_용어지도_부분공간.png | Bin 0 -> 181366 bytes ...장_4가지_기본_부분공간_1785222523974_0.edn | 2 +- Major Study/pages/2.5 미분과 유한차분행렬.md | 8 ++--- Major Study/pages/3.1 벡터공간과 부분공간.md | 19 ++++++----- .../3.2 소거법을 이용한 영공간 계산 A = CR.md | 19 ++++++----- Major Study/pages/3.3 Ax = b의 완전해.md | 28 +++++++++------ Major Study/pages/3.4 독립, 기저, 차원.md | 19 ++++++----- .../pages/3.5 4가지 기본 부분공간의 차원.md | 32 ++++++++++++------ 13 files changed, 79 insertions(+), 50 deletions(-) create mode 100755 Major Study/assets/2장_2.5_격자_경계조건.png create mode 100755 Major Study/assets/2장_2.5_예제_해_비교.png create mode 100755 Major Study/assets/2장_2.5_이차차분_직관.png create mode 100755 Major Study/assets/2장_2.5_차분_세가지.png create mode 100755 Major Study/assets/3장_3.1_용어지도_부분공간.png diff --git a/CLAUDE.md b/CLAUDE.md index 5d08133..d1c3fa5 100755 --- a/CLAUDE.md +++ b/CLAUDE.md @@ -95,6 +95,7 @@ disable-anki-sync:: true - **"왜"를 먼저** — 정의를 던지기 전에 왜 이게 필요한지(무엇이 안 되니까 이걸 만들었는지)부터. 공식은 유도 과정(어디서 태어났는지)과 함께. - **흔한 오해 교정** — 사용자가 헷갈릴 만한 지점을 "~라고 생각하기 쉽지만"으로 명시적으로 짚는다 (2.3 비용, 2.4 부분피벗 보강에서 확립된 패턴). - 사용자가 특정 부분이 이해 안 된다고 하면 그 부분을 위 방식으로 더 잘게 풀어 보강한다 (필요하면 코드·SVG 그림 동원). +- **한 블록에 길게 잇지 말고 블록을 나눌 것** (사용자 요청, 3장 카드화에서 사용자가 일괄 수정한 스타일이 기준): 나열·단계·독립된 문장이 이어지면 한 줄로 쭉 쓰지 말고 **자식 불릿으로 분리**한다. 특히 ❓ 스스로 점검의 답변은 문장·항목 단위로 자식 블록을 나눈다 (카드화 시 cloze·extra 배치가 쉬워짐). 짧은 정의 한 문장은 한 블록이어도 된다. - **문장 중간 '—'(줄표) 구분 금지** (사용자 요청, 1~3장 일괄 제거 완료 — 커밋 d58f8de): 소제목 뒤에는 콜론(`:`), 완결문 뒤에는 마침표로 문장 분리, 부연은 쉼표나 괄호로 연결한다. "📌 원문 — 설명 (p.n)" 형태는 "📌 원문 (설명, p.n)"로 쓴다. 나열형 비교는 '—' 흉내 대신 **진짜 마크다운 표**(`| |` + 구분선 행, 한 블록 안에 연속 줄로)를 쓴다. Claude가 습관적으로 쓰는 문체이므로 작성 후 `grep '—' <새 페이지>` 로 확인한다 (카드 블록 안 기존 '—'는 예외적으로 유지). 정리 사이사이에 다음 요소를 적절히 섞는다 (이모지 마커로 책 내용과 구분): @@ -131,6 +132,7 @@ disable-anki-sync:: true - 작성 규칙: viewBox 약 760px 폭, **흰 배경 rect 포함** (다크 모드 대비), 폰트는 `'Helvetica Neue', Arial, 'Apple SD Gothic Neo', 'Malgun Gothic', sans-serif`, 텍스트 15px 이상. SVG 안에는 LaTeX를 쓸 수 없으니 유니코드 첨자(u₁, h², y″)로 표기한다. - 색 팔레트 (색각이상 검증 통과본): 데이터·개념 선은 파랑 `#1d4ed8` / 주황 `#ea580c` / 초록 `#059669`, 강조 빨강 `#dc2626`, 보조 회색 `#6b7280`/`#9ca3af`, 텍스트 `#111827`/`#374151`. 색만으로 구분하지 말고 범례·직접 라벨을 함께 넣는다. - **커밋 전 시각 검증 필수**: `qlmanage -t -s 1000 -o <스크래치 폴더> <절대경로.svg>` 로 PNG를 만들어 Read 도구로 직접 보고, 좌표 오류(선이 지나야 할 점을 빗나감, 라벨 겹침)를 잡은 뒤 커밋한다. +- ⚠️ **페이지 임베드는 SVG가 아니라 PNG 변환본으로 한다** — Logseq이 `.svg` 임베드를 렌더링하지 못해 빈 블록으로 보인다 (3.1 용어지도에서 실제 발생, 2026-08-06 일괄 교체). 변환: qlmanage 1520px 썸네일 → 정사각 흰 패딩을 viewBox 비율로 중앙 크롭 → `<같은 이름>.png`로 assets에 저장 후 md에서는 `.png`를 참조. SVG 원본은 소스로 assets에 함께 보관한다. - ## 현재 진행: 선형대수학 (스트랭 6판) — `Major Study/` 그래프 - 허브: `스트랭 선형대수학 6판` 페이지. 단원 페이지명은 `1장 벡터와 행렬` 형식이며, 최상단 `deck:: Logseq/전공공부/선형대수학`을 유지한다. - 단원 PDF는 사용자가 스캔해 Logseq에서 링크한다 (예: `assets/1장_벡터와_행렬_*.pdf`). 새 단원 시작 시 사용자가 페이지와 PDF를 세팅한 뒤 정리를 요청한다. 단원 정리는 그 페이지의 PDF 임베드 **아래에** 불릿으로 작성한다. diff --git a/Major Study/assets/2장_2.5_격자_경계조건.png b/Major Study/assets/2장_2.5_격자_경계조건.png new file mode 100755 index 0000000000000000000000000000000000000000..9af642d4bb660560652f9eb1aa9b0d4efae2a67a GIT binary patch literal 51007 zcmeFZc{r5s`v$BP$x=c{K3WK&2-!DkK`@Zh$Ib=rOJ4L>jDi84Wrug zr@AyW=Qe3*PQRr)4L&iS>~W`|X+N*_^s&Bw`r4$Azn;;V_O1Aa{sG-855BOxFZ@pN+^DR@;Hkx_ zw0LOmQgpgi%pHB9MDQGNy$u>kX*_u*xb9p(&GUbEseFB>{=56k`M;0;zs-c4m(h>( z%dGKQnTmx|t^;+}iJfek{>Lrv1jNO~^IY8to+4%sB&Rz_9!I}Z=Mt`v+}fF3EiBB{ z>zVlD(!T^A{dOX`<(?;%7$RP`krTcr3=P?|^HLX~fl01|CE=W_XR;qCeE851G!GA~ z4J4H?u3#nCZ*NR6)&~n1H$%$zM&{-cW(9iX%T-#)2EjsUyO-m1`Z>fb3B4N6OJ%w}K6-3VGexr3Y`&yy*;jm2U>Gn3nd)4 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z8*^w-rs6>6z4|lX3^&Vo5_m0T4vq;1p*#Qz3ojxC zW1iOK62uBs88uudEvkon~DgaMd^b;JzIb#H;=WMuc8 zC>YbUmH`Zh`V#W-!~e}s`!^}-PmgED%>M>{{omZE;OEa=|L=eQ{`JN9|9<+vyYc_x cP7FRgO+#1v2DNak`S)5`NhOIgvCrTBAAP9{rvLx| literal 0 HcmV?d00001 diff --git a/Major Study/assets/3장_4가지_기본_부분공간_1785222523974_0.edn b/Major Study/assets/3장_4가지_기본_부분공간_1785222523974_0.edn index d518daa..f3454ef 100755 --- a/Major Study/assets/3장_4가지_기본_부분공간_1785222523974_0.edn +++ b/Major Study/assets/3장_4가지_기본_부분공간_1785222523974_0.edn @@ -1899,4 +1899,4 @@ :page 65}, :content {:text "[:span]", :image 1785863457636}, :properties {:color "yellow"}}], - :extra {:page 65}} + :extra {:page 2}} diff --git a/Major Study/pages/2.5 미분과 유한차분행렬.md b/Major Study/pages/2.5 미분과 유한차분행렬.md index 2f67003..08fdbec 100755 --- a/Major Study/pages/2.5 미분과 유한차분행렬.md +++ b/Major Study/pages/2.5 미분과 유한차분행렬.md @@ -20,7 +20,7 @@ deck:: Logseq/전공공부/선형대수학 - 후진: $\dfrac{1^2 - 0.9^2}{0.1} = \dfrac{0.19}{0.1} = 1.9$ → 오차 $-0.1$ - 중심: $\dfrac{1.1^2 - 0.9^2}{0.2} = \dfrac{0.4}{0.2} = 2.0$ → **오차 0!** - 💡 보이는가? 전진은 $+0.1$, 후진은 $-0.1$, **정확히 반대 방향으로 같은 크기만큼 틀렸다.** 중심차분은 이 둘의 평균이므로 오차가 서로 지워진다. 중심차분이 좋은 이유는 이게 전부다. - - ![전진·후진·중심차분 비교 (직접 그린 그림)](../assets/2장_2.5_차분_세가지.svg) + - ![전진·후진·중심차분 비교 (직접 그린 그림)](../assets/2장_2.5_차분_세가지.png) - 그림: 전진(파랑)은 너무 가파르고 후진(주황)은 너무 완만한데, 둘을 평균한 중심(초록)은 접선(검정 점선)과 **평행**하다. - **오차에 등급 매기기, 테일러 급수는 "오차 측정기"다** - 위의 "오차 ±0.1"이 우연이 아님을 보장해 주는 도구가 **테일러 급수** (Taylor series)다. 함수를 $x$ 근처에서 전개하면: @@ -41,13 +41,13 @@ deck:: Logseq/전공공부/선형대수학 - 🔢 숫자 확인, $y = x^2$ ($y'' = 2$), $x = 1$, $h = 0.1$: $\dfrac{1.21 - 2 \times 1 + 0.81}{0.01} = \dfrac{0.02}{0.01} = 2$, **정확히 2!** - (이 근사는 중심차분처럼 좌우 대칭이라 3차 다항식까지는 오차가 아예 0이고, 일반 함수에서는 2차 정확도다.) - 💡 또 하나의 직관: **2계도함수 = 양옆 평균과 자신의 차이.** 분자를 다시 묶으면 $y(x+h) + y(x-h) - 2y(x) = 2\left[\frac{y(x+h) + y(x-h)}{2} - y(x)\right]$, "양옆의 평균이 나보다 크면 위로 휘어 있다 ($y'' > 0$)". - - ![2계도함수 = 양옆 평균과 자신의 차이 (직접 그린 그림)](../assets/2장_2.5_이차차분_직관.svg) + - ![2계도함수 = 양옆 평균과 자신의 차이 (직접 그린 그림)](../assets/2장_2.5_이차차분_직관.png) - ⚠️ 1계와 2계는 서로 다른 정보다: "오르막인가"(1계) vs "위로 휘는가"(2계). $y = x^2$ 은 $x < 0$ 에서 내리막이지만 어디서나 위로 휜다 ($y'' = 2 > 0$). - **[3단계] 미분방정식을 격자 위에서 풀기:** $\frac{1}{h^2}KU = F$ **의 탄생** - 풀고 싶은 문제: $-\dfrac{d^2u}{dx^2} = f(x)$ ($0 \le x \le 1$), **경계 조건** $u(0) = 0,\ u(1) = 0$ (양 끝을 벽에 고정). - 💡 왜 $-u''$ 인가 (마이너스가 붙는 이유): 막대의 처짐, 열 분포 같은 물리 문제가 원래 이 부호로 나오기 때문. 이 마이너스 덕분에 행렬에서는 $1, -2, 1$ 이 뒤집혀 $-1, 2, -1$ 패턴이 된다. - **격자 만들기**: 구간 $[0,1]$ 을 $N+1$ 등분 → 간격 $h = \frac{1}{N+1}$. 내부 격자점 $N$ 개에서의 값 $u_1, \cdots, u_N$ 이 **미지수**다. 양 끝 $u_0, u_{N+1}$ 은 미지수가 **아니다**: 경계 조건으로 이미 0이라고 알고 있다. - - ![격자와 경계 조건, N=4 (직접 그린 그림)](../assets/2장_2.5_격자_경계조건.svg) + - ![격자와 경계 조건, N=4 (직접 그린 그림)](../assets/2장_2.5_격자_경계조건.png) - **격자점마다 방정식 하나씩 세운다** ($N = 4$, $h = \frac{1}{5}$ 로 네 개를 전부 써 보면): - $x_1$: $\dfrac{-u_0 + 2u_1 - u_2}{h^2} = f(h)$, 그런데 $u_0 = 0$ → $\dfrac{2u_1 - u_2}{h^2} = f(h)$ - $x_2$: $\dfrac{-u_1 + 2u_2 - u_3}{h^2} = f(2h)$ @@ -61,7 +61,7 @@ deck:: Logseq/전공공부/선형대수학 - 행렬방정식은 $KU = h^2 F = (0.04,\ 0.04,\ 0.04,\ 0.04)$. 소거법으로 풀면: - $$U = (u_1, u_2, u_3, u_4) = (0.08,\ 0.12,\ 0.12,\ 0.08)$$ - 참 해와 비교: $u(0.2) = \frac{0.2 \times 0.8}{2} = 0.08$ ✓, $u(0.4) = \frac{0.4 \times 0.6}{2} = 0.12$ ✓, **완벽히 일치!** (참 해가 2차 다항식이라 오차가 0인 특수한 경우. 일반적인 $f$ 에서는 $h^2$ 에 비례하는 작은 오차가 남고, 격자를 촘촘히 할수록 빠르게 줄어든다.) - - ![행렬로 푼 해와 참 해의 비교 (직접 그린 그림)](../assets/2장_2.5_예제_해_비교.svg) + - ![행렬로 푼 해와 참 해의 비교 (직접 그린 그림)](../assets/2장_2.5_예제_해_비교.png) - 💡 이것이 이 절의 클라이맥스다: **방금 미분방정식을 소거법으로 풀었다.** 컴퓨터가 실제 공학·물리 문제를 푸는 방식이 정확히 이것이고, 그래서 소거법이 "계산 과학에서 가장 비용이 큰 연산"인 것이다. - $K_n$ **의 정확한 형태** 🧩 (고정–고정 경계의 2차차분행렬) - **주대각선은 전부 2, 그 바로 위·아래 대각선은 전부 −1, 나머지는 전부 0** (삼중대각): diff --git a/Major Study/pages/3.1 벡터공간과 부분공간.md b/Major Study/pages/3.1 벡터공간과 부분공간.md index 79ea286..94051c7 100755 --- a/Major Study/pages/3.1 벡터공간과 부분공간.md +++ b/Major Study/pages/3.1 벡터공간과 부분공간.md @@ -3,7 +3,7 @@ deck:: Logseq/전공공부/선형대수학 - ### 핵심 개념 - **이 절의 목표**: 지금까지 벡터를 "숫자 묶음"으로만 다뤘다면, 이제 벡터들이 사는 **"세계"(공간)** 자체를 다룬다. 이 절에서 용어 4개(벡터공간·부분집합·부분공간·생성)를 잡아두면, 3장 전체(열공간·영공간·기저·차원)가 이 위에 쌓인다. - **[1단계] 용어부터 확실히, 벡터공간 vs 부분집합 vs 부분공간** (혼동 주의 ⚠️) - - ![용어 지도: 부분집합·부분공간·행렬이 만드는 공간들 (직접 그린 그림)](../assets/3장_3.1_용어지도_부분공간.svg) + - ![용어 지도: 부분집합·부분공간·행렬이 만드는 공간들 (직접 그린 그림)](../assets/3장_3.1_용어지도_부분공간.png) - **벡터공간** (vector space): 벡터들의 "세계". **더하기(** $v+w$ **)와 스칼라곱(** $cv$ **)을 아무리 해도 그 안에서 벗어나지 않는** 벡터들의 집합이다. 대표: $\mathbb{R}^1, \mathbb{R}^2, \mathbb{R}^3, \cdots$ - 📌 원문 (p.132): ((6a685a78-846c-4c02-976a-445b78c8c392)) - 정확히는 8가지 조건(교환법칙, 결합법칙, 영벡터 존재 등)을 만족해야 한다. 3.1 연습문제 서두에 목록이 있고, 문제 1~3이 이 조건들을 갖고 논다. $\mathbb{R}^n$ 에서는 자명하게 성립하므로 **외울 필요는 없다**. 핵심 한 줄만: **모든 선형결합** $cv + dw$ **가 그 안에 있어야 한다.** @@ -71,21 +71,22 @@ deck:: Logseq/전공공부/선형대수학 - 🧭 **이정표**: 이 절부터 "선형독립", "생성", "기저", "차원" 같은 말이 정의 없이 계속 나온다. **정식 정의는 전부 [[3.4 독립, 기저, 차원]] 에 있다.** 그 전까지는 감(독립 ≈ 서로 겹치지 않음, 기저 ≈ 공간을 만드는 최소 재료)으로 읽고 넘어가면 된다. - 📌 원문 (기저 예고, p.132): ((6a685b10-625e-4ab7-a79f-76f40b0f0ba6)) / ((6a685b18-d976-44bd-b7b6-d36e4770223b)) - ### ❓ 스스로 점검 - - 부분집합과 부분공간의 차이를 한 문장으로 말하라. + - 부분집합과 부분공간의 차이를 한 문장으로 말하라. #card - 부분집합은 벡터들을 그냥 골라 담은 것(조건 없음), 부분공간은 그중 $v+w$ 와 $cv$ 에 닫혀 있어서 그 자체로 벡터공간이 되는 것. - - 부분공간 판정법 2가지는? 이것만 확인하면 8가지 조건은 왜 자동인가? - - ① $v + w$ 가 안에 있다 ② $cv$ 가 안에 있다. 연산 자체를 원래 벡터공간에서 물려받으므로 교환·결합법칙 등은 자동 성립. - - 왜 모든 부분공간은 영벡터를 포함해야 하는가? 이 사실로 즉시 탈락시킬 수 있는 집합의 예는? + - 부분공간 판정법 2가지는? 이것만 확인하면 8가지 조건은 왜 자동인가? #card + - ① $v + w$ 가 안에 있다 ② $cv$ 가 안에 있다. + - 연산 자체를 원래 벡터공간에서 물려받으므로 교환·결합법칙 등은 자동 성립. + - 왜 모든 부분공간은 영벡터를 포함해야 하는가? 이 사실로 즉시 탈락시킬 수 있는 집합의 예는? #card - 판정법 ②에 $c = 0$ 을 넣으면 $0v = 0$ 이 포함되어야 하기 때문. 원점을 지나지 않는 직선·평면은 즉시 탈락. - - $\mathbb{R}^3$ 의 부분공간을 전부 나열하라. + - $\mathbb{R}^3$ 의 부분공간을 전부 나열하라. #card - $\mathbf{Z}$ (영벡터만), 원점 지나는 직선 $\mathbf{L}$, 원점 지나는 평면 $\mathbf{P}$, $\mathbb{R}^3$ 전체. - - 제1사분면은 어느 판정 조건에서 탈락하는가? 제1∪제3사분면은? + - 제1사분면은 어느 판정 조건에서 탈락하는가? 제1∪제3사분면은? #card - 제1사분면: 스칼라곱($c = -1$)에서 탈락. 제1∪제3사분면: 덧셈에서 탈락 (예: $(2,3)+(-3,-2) = (-1,1)$). - 열공간 $\mathbf{C}(A)$ 의 정의는? $Ax = b$ 의 해가 존재할 필요충분조건을 열공간의 언어로 말하라. - $A$ 의 열들의 모든 선형결합(= 가능한 모든 $Ax$)의 집합. 해 존재 ⟺ $b$ 가 $\mathbf{C}(A)$ 안에 있다. - "$A$ 의 열들의 집합"은 왜 부분공간이 아닌가? 가역행렬 전체의 집합은 왜 부분공간이 아닌가? - 열들의 집합은 유한집합이라 선형결합이 빠져 있다 (생성해야 열공간이 됨). 가역행렬 집합은 영행렬이 없고, $I + (-I) = 0$ 처럼 덧셈에 안 닫힌다. - - 행공간을 전치행렬로 표현하면? $m \times n$ 행렬의 열공간과 행공간은 각각 어디의 부분공간인가? + - 행공간을 전치행렬로 표현하면? $m \times n$ 행렬의 열공간과 행공간은 각각 어디의 부분공간인가? #card - 행공간 $= \mathbf{C}(A^{\mathrm{T}})$. 열공간은 $\mathbb{R}^m$, 행공간은 $\mathbb{R}^n$ 의 부분공간. - ### ✏️ 연습문제 풀이 (솔루션 참고) - ※ 문제 1~7은 "일반적인 벡터공간"(꼭 열벡터가 아닌)에 관한 문제다. 벡터공간의 8가지 조건 목록은 책 p.139 참고, (1)~(4)는 덧셈, (5)~(6)은 스칼라곱, (7)~(8)은 둘의 연결. @@ -131,4 +132,4 @@ deck:: Logseq/전공공부/선형대수학 - **문제 14**: (a) $\mathbf{M}$ 에서 가역행렬의 집합이 부분공간이 아님을 보여라. (b) 특이행렬의 집합이 부분공간이 아님을 보여라. - (a) 영행렬이 없다 (영행렬은 비가역). 또 $I + (-I) = 0$, 가역 둘의 합이 비가역. - (b) $\begin{bmatrix} 1&0 \\ 0&0 \end{bmatrix} + \begin{bmatrix} 0&0 \\ 0&1 \end{bmatrix} = I$, 특이행렬 둘의 합이 가역. 덧셈에 안 닫힘. - - 💡 본문 [4단계]의 "가역·특이행렬은 부분공간을 형성하지 않는다" 하이라이트가 바로 이 문제다. + - 💡 본문 [4단계]의 "가역·특이행렬은 부분공간을 형성하지 않는다" 하이라이트가 바로 이 문제다. \ No newline at end of file diff --git a/Major Study/pages/3.2 소거법을 이용한 영공간 계산 A = CR.md b/Major Study/pages/3.2 소거법을 이용한 영공간 계산 A = CR.md index 9c3677a..5880d77 100755 --- a/Major Study/pages/3.2 소거법을 이용한 영공간 계산 A = CR.md +++ b/Major Study/pages/3.2 소거법을 이용한 영공간 계산 A = CR.md @@ -72,19 +72,22 @@ deck:: Logseq/전공공부/선형대수학 - 📌 원문 (p.151): ((6a6b6080-45c0-4450-b783-ca74d3bbf525)) - 💡 영공간의 크기 감각: 영공간의 '차원' = 자유변수 개수 $(n-r)$. (차원의 정식 정의는 [[3.4 독립, 기저, 차원]], 4공간 총정리는 [[3.5 4가지 기본 부분공간의 차원]].) - ### ❓ 스스로 점검 - - 자유변수란 무엇인가? $R_0$ 만 보고 어떤 변수가 자유변수인지 판정하는 법은? - - 소거 후에도 방정식이 붙잡지 않아 아무 값이나 넣어도 되는 미지수. 판정: 각 행의 선두 1(피벗)이 있는 열의 변수는 피벗변수, 피벗이 없는 열의 변수는 자유변수. - - 특별해는 어떻게 만드는가? 몇 개가 나오며, 영공간과의 관계는? - - 자유변수 하나에 1, 나머지 자유변수에 0을 넣고 피벗변수를 역대입으로 구한다. $(n-r)$ 개가 나오고, 영공간 = 특별해들의 모든 선형결합 (특별해가 영공간의 기저). + - 자유변수란 무엇인가? $R_0$ 만 보고 어떤 변수가 자유변수인지 판정하는 법은? #card + - 소거 후에도 방정식이 붙잡지 않아 아무 값이나 넣어도 되는 미지수. + - 판정: 각 행의 선두 1(피벗)이 있는 열의 변수는 피벗변수, 피벗이 없는 열의 변수는 자유변수. + - 특별해는 어떻게 만드는가? 몇 개가 나오며, 영공간과의 관계는? #card + - 자유변수 하나에 1, 나머지 자유변수에 0을 넣고 피벗변수를 역대입으로 구한다. + - $(n-r)$ 개가 나오고, 영공간 = 특별해들의 모든 선형결합 (특별해가 영공간의 기저). - $[\,I\ \ F\,]x = 0$ 의 특별해를 행렬 하나로 표현하면? 검산은 왜 자동인가? - $\begin{bmatrix} -F \\ I \end{bmatrix}$ 의 열들 (열 섞였으면 $P^{\mathrm{T}}$ 를 앞에). $[\,I\ F\,]\begin{bmatrix} -F \\ I \end{bmatrix} = -F + F = 0$ 이므로. - $H = WF$ 에서 $W, H, F$ 는 각각 무엇인가? 책 p.147 박스의 오타는 무엇인가? - $W$ = 독립인 열들, $H$ = 종속인 열들, $F$ = 독립을 결합해 종속을 만드는 계수(조리법). 박스는 "(독립 열) = (종속 열)×F"로 뒤바뀌어 인쇄됨. 올바른 방향은 (종속 열) = (독립 열)×F. - "$R_0$ 안의 $I$ 가 $C$ 의 위치를 알려준다"에서 '위치'란? $C$ 의 열은 어디서 가져오는가? - $I$ 를 이루는 피벗 열들의 열 번호. $C$ 의 열은 $R_0$ 가 아니라 $A$ **의 원본 열**을 그 번호대로 뽑아온다. - - 왜 $\mathbf{N}(A) = \mathbf{N}(R_0)$ 인가? - - 소거 = 가역행렬 $E$ 를 곱하는 것. $Ax=0$ 이면 $EAx=0$; 역으로 $EAx=0$ 이면 $E^{-1}$ 를 곱해 $Ax=0$. 해집합이 같다. - - $n > m$ 이면 $Ax = 0$ 에 0 아닌 해가 반드시 존재하는 이유를 한 줄로. + - 왜 $\mathbf{N}(A) = \mathbf{N}(R_0)$ 인가? #card + - 소거 = 가역행렬 $E$ 를 곱하는 것. $Ax=0$ 이면 $R_0 = EAx=0$ + - 역으로 $EAx=0$ 이면 $E^{-1}$ 를 곱해 $Ax=0$. 해집합이 같다. + - $n > m$ 이면 $Ax = 0$ 에 0 아닌 해가 반드시 존재하는 이유를 한 줄로. #card - 피벗 ≤ $m$ < $n$ → 자유변수 ≥ 1 → 특별해 존재. - ### ✏️ 연습문제 풀이 (솔루션 참고) - **문제 1** (카드화하기 좋은 문제): $E$ 가 가역행렬일 때, $A$ 와 $R = EA$ 는 왜 동일한 영공간을 갖는가? @@ -110,4 +113,4 @@ deck:: Logseq/전공공부/선형대수학 - **문제 13** (꼭 풀어볼 만한 문제): 평면 $x - 3y - z = 12$ 는 $x - 3y - z = 0$ 과 평행하다. $(12, 0, 0)$ 이 이 평면 위에 있을 때, 평면 위의 모든 점을 다음 꼴로 나타내라: $\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} \square \\ 0 \\ 0 \end{bmatrix} + y\begin{bmatrix} \square \\ 1 \\ 0 \end{bmatrix} + z\begin{bmatrix} \square \\ 0 \\ 1 \end{bmatrix}$ - $$\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 12 \\ 0 \\ 0 \end{bmatrix} + y\begin{bmatrix} 3 \\ 1 \\ 0 \end{bmatrix} + z\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}$$ - 검산: $x = 12 + 3y + z$ → $x - 3y - z = 12$ ✓. - - 💡 구조를 보라: (평면 위 한 점) + (평행한 원점 평면의 모든 점): 다음 절 완전해 $x = x_p + x_n$ 의 **예고편**이다. $y, z$ 가 자유변수, $(3,1,0)$ 과 $(1,0,1)$ 이 특별해 역할. + - 💡 구조를 보라: (평면 위 한 점) + (평행한 원점 평면의 모든 점): 다음 절 완전해 $x = x_p + x_n$ 의 **예고편**이다. $y, z$ 가 자유변수, $(3,1,0)$ 과 $(1,0,1)$ 이 특별해 역할. \ No newline at end of file diff --git a/Major Study/pages/3.3 Ax = b의 완전해.md b/Major Study/pages/3.3 Ax = b의 완전해.md index 58d449a..92a10f4 100755 --- a/Major Study/pages/3.3 Ax = b의 완전해.md +++ b/Major Study/pages/3.3 Ax = b의 완전해.md @@ -69,20 +69,28 @@ deck:: Logseq/전공공부/선형대수학 - (읽을거리) **랜덤 선형대수학**: 행렬이 너무 크면 열을 **샘플링**해서 $A \approx CUR$ 로 근사한다. 오차를 감수하고 속도를 얻는 현대적 기법. "선형대수학은 살아있다." - 📌 원문 (p.165): ((6a6c91c8-cec5-4d43-a8f1-c4f9cf27cc20)) - ### ❓ 스스로 점검 - - 특수해·특별해·완전해를 각각 한 문장으로 정의하라. - - 특수해 $x_p$: $Ax=b$ 의 해 하나 (자유변수 = 0 으로 고른 것). 특별해 $s$: $Ax=0$ 의 해 (자유변수 하나만 1). 완전해: $x = x_p + c_1 s_1 + \cdots + c_{n-r}s_{n-r}$, $Ax=b$ 의 모든 해. - - 완전해가 "특수해 + 영공간"인 이유를 두 방향으로 증명하라. - - ① $A(x_p + x_n) = b + 0 = b$ → 이 꼴은 전부 해. ② 임의의 해 $x$ 에 대해 $A(x - x_p) = b - b = 0$ → $x - x_p \in \mathbf{N}(A)$ → 모든 해가 이 꼴. + - 특수해·특별해·완전해를 각각 한 문장으로 정의하라. #card + - 특수해 $x_p$: $Ax=b$ 의 해 하나 (자유변수 = 0 으로 고른 것). + - 특별해 $s$: $Ax=0$ 의 해 (자유변수 하나만 1). + - 완전해: $x = x_p + c_1 s_1 + \cdots + c_{n-r}s_{n-r}$, $Ax=b$ 의 모든 해. + - 완전해가 "특수해 + 영공간"인 이유를 두 방향으로 증명하라. #card + - ① $A(x_p + x_n) = b + 0 = b$ → 이 꼴은 전부 해. + - ② 임의의 해 $x$ 에 대해 $A(x - x_p) = b - b = 0$ → $x - x_p \in \mathbf{N}(A)$ → 모든 해가 이 꼴. - $Ax = b$ 의 해가 존재할 조건을 $[\,R_0\ \ d\,]$ 의 언어로 말하라. - $R_0$ 의 영행이 있는 위치에서 $d$ 의 성분도 0이어야 한다 ($0 = 0$). 하나라도 $0 \neq d_i$ 면 해 없음. - 특수해에서 자유변수를 0으로 놓는 이유는? 그때 피벗변수는 어디서 바로 읽는가? - 가장 간단해서, 그리고 $R_0$ 의 피벗 부분이 $I$ 라서 피벗변수 값이 우변 $d$ 에 그대로 나타난다. - $r = n$ 일 때와 $r = m$ 일 때 해의 개수는 각각 어떻게 되는가? - - $r = n$ (최대 열 랭크): 0 또는 1개 (영공간 $\{0\}$, 존재는 $b$ 조건부). $r = m$ (최대 행 랭크): 항상 존재, $n = m$ 이면 1개, $n > m$ 이면 ∞개. - - 특수해는 왜 상수배를 하면 안 되는가? - - $A(2x_p) = 2b \neq b$. 상수배·결합이 자유로운 것은 영공간 부분 $x_n$ 뿐이다. - - 랭크 $r$, 크기 $m \times n$ 에 따른 해의 개수 4가지 유형을 표로 재현하라. - - $r=m=n$: 1개 / $r=m m$ 이면 ∞개. + - 특수해는 왜 상수배를 하면 안 되는가? #card + - $A(2x_p) = 2b \neq b$. + extra:: 상수배·결합이 자유로운 것은 영공간 부분 $x_n$ 뿐이다. + - 랭크 $r$, 크기 $m \times n$ 에 따른 해의 개수 4가지 유형을 표로 재현하라. #card + - $r=m=n$ : 1개 + - $r=m m$. 열로 세우면 가로로 긴 행렬 → 피벗 ≤ m < n → 자유변수 존재 → 0 아닌 결합이 영벡터를 만든다. - 기저의 정의 두 조건은? 각각이 "표현의 존재"와 "표현의 유일성" 중 무엇을 책임지는가? - - ① 선형독립 ② 공간 생성. 생성 → 모든 벡터가 기저의 결합으로 표현됨(존재), 독립 → 그 표현이 유일함. + - ① 선형독립 + - ② 공간 생성. + - 생성 → 모든 벡터가 기저의 결합으로 표현됨(존재), 독립 → 그 표현이 유일함. - "기저 표현이 유일하다"의 증명을 재현하라. - $v = \sum a_i v_i = \sum b_i v_i$ 라면 빼서 $\sum (a_i - b_i)v_i = 0$. 기저는 독립이므로 계수 전부 0 → $a_i = b_i$. - - 가역행렬의 열들이 $\mathbb{R}^n$ 의 기저인 이유를 두 조건으로 나눠 설명하라. - - 독립: $Ax=0 \Rightarrow x = A^{-1}0 = 0$. 생성: 임의의 $b$ 에 대해 $x = A^{-1}b$ 가 존재해 $Ax = b$. + - 가역행렬의 열들이 $\mathbb{R}^n$ 의 기저인 이유를 두 조건으로 나눠 설명하라. #card + - 독립: $Ax=0 \Rightarrow x = A^{-1}0 = 0$. + - 생성: 임의의 $b$ 에 대해 $x = A^{-1}b$ 가 존재해 $Ax = b$. - 차원의 정의는? "모든 기저는 같은 개수"의 증명 아이디어는? - - 차원 = 모든 기저에 있는 벡터의 개수. 기저가 $m$ 개, $n$ 개($n>m$)라면 $W = VA$ 에서 $A$ 가 가로로 긴 행렬 → $Ax=0$ 비자명해 → $W$ 종속 → 모순. + - 차원 = 모든 기저에 있는 벡터의 개수. + - 기저가 $m$ 개, $n$ 개($n>m$)라면 $W = VA$ 에서 $A$ 가 가로로 긴 행렬 → $Ax=0$ 비자명해 → $W$ 종속 → 모순. - 다음 용어 조합의 옳고 그름: 공간의 랭크 / 행렬의 랭크 / 기저의 차원 / 열공간의 차원 / 행렬의 기저. - 행렬의 랭크 ○, 열공간의 차원 ○ (그리고 이 둘은 같은 수). 공간의 랭크 ✗, 기저의 차원 ✗, 행렬의 기저 ✗. - - 2×2 대칭행렬 공간의 기저와 차원은? $y'' = -y$ 의 해공간의 기저와 차원은? - - $A_1, A_4, A_2+A_3$ → 3차원. $\{\sin x, \cos x\}$ → 2차원. - $y'' = 2$ 의 해집합은 왜 벡터공간이 아닌가? 해집합의 구조는? - - 0 함수가 해가 아니라서 (부분공간은 0 포함 필수). 구조는 특수해 + 영공간: $y = x^2 + cx + d$, $Ax=b$ 완전해와 동일. + - 0 함수가 해가 아니라서 (부분공간은 0 포함 필수). + - 구조는 특수해 + 영공간: $y = x^2 + cx + d$, $Ax=b$ 완전해와 동일. - ### ✏️ 연습문제 풀이 (솔루션 참고) - **문제 2** (꼭 풀어볼 만한 문제): 6개의 벡터 $v_1 = (1,-1,0,0)$, $v_2 = (1,0,-1,0)$, $v_3 = (1,0,0,-1)$, $v_4 = (0,1,-1,0)$, $v_5 = (0,1,0,-1)$, $v_6 = (0,0,1,-1)$ 중 선형독립인 벡터를 최대한 많이 찾아라. - 답: **최대 3개** (예: $v_1, v_2, v_3$). @@ -120,4 +123,4 @@ deck:: Logseq/전공공부/선형대수학 - (b) $(1,-1,0,0), (1,0,-1,0), (1,0,0,-1)$ → 3차원. (조건식 1개 = $[\,1\ 1\ 1\ 1\,]x = 0$ 의 영공간, $n - r = 4 - 1 = 3$.) - (c) 두 조건 = $2 \times 4$ 행렬의 영공간, 차원 $4 - 2 = 2$: 기저 $(1,-1,-1,0), (1,-1,0,-1)$. - (d) $\mathbf{C}(I) = \mathbb{R}^4$ 의 기저 = $I$ 의 네 열. $\mathbf{N}(I) = \mathbf{Z}$ 의 기저 = **공집합** (관례). - - 💡 (b)(c)에서 기저 만드는 실전 요령: 조건식을 행렬로 쓰고 **영공간의 특별해**를 구하면 그것이 기저다. (기저는 유일하지 않다. 답이 달라도 개수만 맞으면 된다.) + - 💡 (b)(c)에서 기저 만드는 실전 요령: 조건식을 행렬로 쓰고 **영공간의 특별해**를 구하면 그것이 기저다. (기저는 유일하지 않다. 답이 달라도 개수만 맞으면 된다.) \ No newline at end of file diff --git a/Major Study/pages/3.5 4가지 기본 부분공간의 차원.md b/Major Study/pages/3.5 4가지 기본 부분공간의 차원.md index 28f2347..bcbae75 100755 --- a/Major Study/pages/3.5 4가지 기본 부분공간의 차원.md +++ b/Major Study/pages/3.5 4가지 기본 부분공간의 차원.md @@ -50,20 +50,32 @@ deck:: Logseq/전공공부/선형대수학 - $$A = CR = u_1 v_1^{\mathrm{T}} + u_2 v_2^{\mathrm{T}} + \cdots + u_r v_r^{\mathrm{T}} \qquad (u_i = C\text{의 열},\ v_i^{\mathrm{T}} = R\text{의 행})$$ - 💡 "모든 행렬은 랭크 1 조각 $r$ 개로 분해된다". 뒤에서 배울 특잇값 분해(SVD)의 원형이다. - ### ❓ 스스로 점검 - - 4가지 기본 부분공간의 이름·정의·사는 곳·차원을 표로 재현하라. - - 행공간 $\mathbf{C}(A^{\mathrm{T}})$/행들의 결합/$\mathbb{R}^n$/$r$ · 열공간 $\mathbf{C}(A)$/열들의 결합/$\mathbb{R}^m$/$r$ · 영공간 $\mathbf{N}(A)$/$Ax=0$ 해/$\mathbb{R}^n$/$n-r$ · 좌영공간 $\mathbf{N}(A^{\mathrm{T}})$/$A^{\mathrm{T}}y=0$ 해/$\mathbb{R}^m$/$m-r$. - - $R_0$ 에서 네 공간의 기저를 각각 어떻게 읽는가? - - 행공간: 영행 아닌 $r$ 개 행 / 열공간: 피벗 열 (단, $A$ **의** 원본 피벗 열) / 영공간: 특별해 $(n-r)$ 개 / 좌영공간: $R_0^{\mathrm{T}}y = 0$ 의 해, $A$ 기준으로는 $[\,A\ I\,] \to [\,R_0\ E\,]$ 에서 $E$ 의 마지막 $(m-r)$ 행. - - 소거에서 보존되는 공간과 변하는 공간은? "$A$ 와 $R_0$ 의 열공간이 같다"가 거짓인 이유는? - - 보존: 행공간·영공간. 변함: 열공간·좌영공간 (차원만 보존). $R_0$ 의 열은 영행 때문에 끝이 0이지만 $A$ 의 열은 아니라서. + - 4가지 기본 부분공간의 이름·정의·사는 곳·차원을 표로 재현하라. #card + - 행공간 $\mathbf{C}(A^{\mathrm{T}})$ / 행들의 결합 / $\mathbb{R}^n$ / $r$ + - 열공간 $\mathbf{C}(A)$ / 열들의 결합 / $\mathbb{R}^m$ / $r$ + - 영공간 $\mathbf{N}(A)$ / $Ax=0$ 해 / $\mathbb{R}^n$ / $n-r$ + - 좌영공간 $\mathbf{N}(A^{\mathrm{T}})$ / $A^{\mathrm{T}}y=0$ 해 / $\mathbb{R}^m$ / $m-r$. + - $R_0$ 에서 네 공간의 기저를 각각 어떻게 읽는가? #card + - 행공간: 영행 아닌 $r$ 개 행 + - 열공간: 피벗 열 (단, $A$ **의** 원본 피벗 열) + - 영공간: 특별해 $(n-r)$ 개 + - 좌영공간: $R_0^{\mathrm{T}}y = 0$ 의 해, $A$ 기준으로는 $[\,A\ I\,] \to [\,R_0\ E\,]$ 에서 $E$ 의 마지막 $(m-r)$ 행. + - 소거에서 보존되는 공간과 변하는 공간은? "$A$ 와 $R_0$ 의 열공간이 같다"가 거짓인 이유는? #card + - 보존: 행공간·영공간. + - 변함: 열공간·좌영공간 (차원만 보존). $R_0$ 의 열은 영행 때문에 끝이 0이지만 $A$ 의 열은 아니라서. - "독립인 열의 개수 = 독립인 행의 개수"가 성립하는 이유를 한 줄로. - 피벗 하나가 피벗 행과 피벗 열을 동시에 하나씩 만들기 때문 (둘 다 = 피벗 개수 $r$). - 카운팅 정리 두 개를 쓰고 의미를 말하라. - - $r + (n-r) = n$: $\mathbb{R}^n$ 을 행공간 차원 + 영공간 차원이 채운다. $r + (m-r) = m$: $\mathbb{R}^m$ 을 열공간 + 좌영공간이 채운다. + - $r + (n-r) = n$ : $\mathbb{R}^n$ 을 행공간 차원 + 영공간 차원이 채운다. + - $r + (m-r) = m$ : $\mathbb{R}^m$ 을 열공간 + 좌영공간이 채운다. - 좌영공간이라는 이름의 유래는? - $A^{\mathrm{T}}y = 0$ ⟺ $y^{\mathrm{T}}A = 0^{\mathrm{T}}$, $y$ 가 $A$ 의 왼쪽에 곱해지는 영공간. - - $7 \times 9$ 행렬의 랭크가 5일 때 네 공간의 차원은? - - 행공간 5, 열공간 5, 영공간 $9-5=4$, 좌영공간 $7-5=2$. (합 = $16 = m + n$) + - $7 \times 9$ 행렬의 랭크가 5일 때 네 공간의 차원은? #card + - 행공간 5 + - 열공간 5 + - 영공간 $9-5=4$ + - 좌영공간 $7-5=2$ + - (합 = $16 = m + n$) - ### ✏️ 연습문제 풀이 (솔루션 참고) - **문제 1**: (a) $7 \times 9$ 행렬의 랭크가 5일 때 4공간의 차원과 그 합은? (b) $3 \times 4$ 행렬의 랭크가 3일 때 열공간과 좌영공간은? - (a) $5, 5, 4, 2$, 합 $= 16 = m + n$. (b) 열공간 = $\mathbb{R}^3$ 전체 (최대 행 랭크), 좌영공간 = $\{0\}$ (차원 0). @@ -81,4 +93,4 @@ deck:: Logseq/전공공부/선형대수학 - **문제 24** (중요한 문제): 벡터 $d$ 가 4공간 중 어디에 있으면 $A^{\mathrm{T}}y = d$ 의 해가 존재하는가? 어떤 공간이 영벡터만 포함하면 그 해가 유일한가? - 존재: $d \in \mathbf{C}(A^{\mathrm{T}})$, 즉 $d$ 가 $A$ **의 행공간**에 있을 때. ($A^{\mathrm{T}}$ 의 열공간 = $A$ 의 행공간이므로 3.1의 "해 존재 ⟺ 우변이 열공간에" 를 $A^{\mathrm{T}}$ 에 적용한 것.) - 유일: $\mathbf{N}(A^{\mathrm{T}})$ (좌영공간) $= \{0\}$ 일 때. - - 💡 모든 해 존재/유일성 문제는 "열공간에 있나 / 영공간이 0뿐인가"로 환원된다. 대상이 $A^{\mathrm{T}}$ 면 행공간/좌영공간으로 이름만 바뀔 뿐. + - 💡 모든 해 존재/유일성 문제는 "열공간에 있나 / 영공간이 0뿐인가"로 환원된다. 대상이 $A^{\mathrm{T}}$ 면 행공간/좌영공간으로 이름만 바뀔 뿐. \ No newline at end of file