From 451457ba87bff0745388b9ef1502cf60cd431768 Mon Sep 17 00:00:00 2001 From: songyc macbook Date: Mon, 31 Aug 2026 00:24:38 +0900 Subject: [PATCH] =?UTF-8?q?5.2=20=ED=96=89=EB=A0=AC=EC=8B=9D=EC=9D=98=20?= =?UTF-8?q?=EA=B3=84=EC=82=B0=EA=B3=BC=20=EC=9D=91=EC=9A=A9=20=EC=A0=95?= =?UTF-8?q?=EB=A6=AC:=208=EB=8B=A8=EA=B3=84=20=EA=B5=AC=EC=A1=B0(=EA=B3=A0?= =?UTF-8?q?=EC=9C=B3=EA=B0=92=20=EC=88=98=EC=9C=84=EC=A1=B0=EC=A0=88=C2=B7?= =?UTF-8?q?=EC=8B=9D6=20=EC=83=81=EC=84=B8=C2=B7detAB=20=EC=A6=9D=EB=AA=85?= =?UTF-8?q?=203=EC=88=98=C2=B7EX1/EX2=C2=B7=ED=81=B0=20=EA=B3=B5=EC=8B=9D?= =?UTF-8?q?=20SVG),=20=ED=95=98=EC=9D=B4=EB=9D=BC=EC=9D=B4=ED=8A=B8=2031?= =?UTF-8?q?=EA=B1=B4=20=EC=97=B0=EA=B2=B0=C2=B7OCR=20=EA=B5=90=EC=A0=95=20?= =?UTF-8?q?15=EA=B1=B4,=20=EC=97=B0=EC=8A=B5=EB=AC=B8=EC=A0=9C=208?= =?UTF-8?q?=EC=A0=9C(1=C2=B72=C2=B73=C2=B75=C2=B77=C2=B710=C2=B713=C2=B716?= =?UTF-8?q?)=20=ED=95=B4=EC=84=A4,=20=EC=8A=A4=EC=8A=A4=EB=A1=9C=20?= =?UTF-8?q?=EC=A0=90=EA=B2=80=208=EB=AC=B8=20=EC=B9=B4=EB=93=9C=ED=99=94(?= =?UTF-8?q?=EC=82=AC=EC=9A=A9=EC=9E=90),=20CLAUDE.md=20=EC=A7=84=ED=96=89?= =?UTF-8?q?=20=EC=83=81=ED=99=A9=20=EA=B0=B1=EC=8B=A0?= 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_018JYaHeD41WPAk5P5wBojQe --- CLAUDE.md | 2 +- Major Study/assets/5장_5.2_큰공식_치환6개.png | Bin 0 -> 117073 bytes Major Study/assets/5장_5.2_큰공식_치환6개.svg | 144 +++++ .../assets/5장_p289_두행렬의곱_식3.png | Bin 0 -> 64658 bytes .../assets/5장_p289_삼각대각행렬_식1.png | Bin 0 -> 80922 bytes Major Study/assets/5장_p289_전치행렬_식2.png | Bin 0 -> 68262 bytes .../assets/5장_p290_기본성질3가지_식5.png | Bin 0 -> 434350 bytes .../assets/5장_p290_피벗들의곱_식4.png | Bin 0 -> 85100 bytes .../assets/5장_p291_detAB증명_성질확인.png | Bin 0 -> 281868 bytes .../5장_p291_크레이머_핵심아이디어_식8.png | Bin 0 -> 112198 bytes .../5장_p291_행교환없을때_피벗곱_식7.png | Bin 0 -> 100696 bytes 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--git a/CLAUDE.md b/CLAUDE.md index c79fcdd..58fd388 100755 --- a/CLAUDE.md +++ b/CLAUDE.md @@ -149,7 +149,7 @@ disable-anki-sync:: true - **해답집**: `Major Study/assets/solution_1784335983942_0.pdf` — 스트랭 6판 공식 영문 솔루션 (전 챕터). 문제 번호는 한국어판과 동일하다. 각 챕터의 "Problem Set x.y" 헤더로 위치를 찾는다 (1장 해답: PDF 2~18쪽). - **PDF 읽는 법**: 단원 PDF는 스캔+OCR본이라 텍스트 레이어의 수식이 깨져 있다. 정리할 때는 **Read 도구로 페이지를 시각적으로 읽는다** (pdftoppm/poppler 설치됨). 이미지 추출·edn 좌표 파싱은 PyMuPDF(`import fitz`, 설치됨)를 쓴다. - **쪽번호 오프셋**: 1장 PDF는 책 쪽번호 = PDF 쪽번호 + 18. 2장 PDF는 책 쪽번호 = PDF 쪽번호 + 70. 단원 PDF마다 오프셋이 다를 수 있으니 첫 몇 쪽으로 확인할 것. -- **진행 상황 (2026-08-30 기준)**: **1~4장 완전 종료** (정리·카드화·「이 단원 핵심문제」 복습까지, 2026-08-29 확인. 4장 GoodNotes PDF 삭제 완료). 그래프 3분할·Anki 프로파일 3분할·모델명 이전·AnkiWeb 계정 분리·모바일 Working Copy 전환 모두 완결. **현재: 5장 행렬식 진행 중 (소단원별 사이클 + GoodNotes 플로우)**. 5.1 「3×3 행렬식과 여인수」 정리 완료: 5장 페이지(서론+소단원 목록) + 5.1 페이지, 하이라이트 21개(2차 4개 포함) 연결, PDF 추출 이미지 10개 + 자작 SVG 1개(식 3 분해), ACᵀ 비대각성분 상세 보강(어긋난 짝짓기 → A* 치환 → 0), 연습문제 7제(1·3·5·6·13·16·18) 해설. 사용자 카드화 일부 진행(#card 3장, 추가 예정), **현재 단계: 사용자가 5.1 연습문제를 풀며 질문·해설 보강 요청 예정** (질문에 "수정하지 말고 답변만"이 붙으면 파일을 건드리지 않는다). 5장 쪽번호 오프셋: 책 쪽 = PDF 쪽 + 278 (총 24쪽, 5.1은 책 281~288·연습문제는 PDF 8~10쪽). 해답집 Problem Set 5.1은 해답 PDF 83~84쪽. ⚠️ 한국어판 5.1 연습문제 6의 행렬 A(주대각 1)가 원서(본문 예제 A₃)와 달라 해답집 값 detA=4와 불일치, 한국어판 기준 정답은 −1 (해설에 주석 완료. 이후 장에서도 해답집 검산 불일치 시 한국어판 행렬 원문부터 재확인할 것). 남은 뒷정리: 사용자 Anki 싱크 시 Major Study 프로파일·create 수 검증. 다음: 5.2 (사이클 계속). 2장에서 확립된 방식(🧩 행렬 형태 명시·숫자 예제 먼저·단계 지도·흔한 오해 교정·자작 SVG) 계속 적용. 이해 안 되는 부분 요청 시 해당 블록만 보강하는 패턴 유지. +- **진행 상황 (2026-08-31 기준)**: **1~4장 완전 종료** (정리·카드화·「이 단원 핵심문제」 복습까지). 그래프 3분할·Anki 프로파일 3분할·모델명 이전·AnkiWeb 계정 분리·모바일 Working Copy 전환 모두 완결. **현재: 5장 행렬식 진행 중 (소단원별 사이클)**. **5.1 완전 종료** (정리·연습문제 7제(1·3·5·6·13·16·18)·사용자 풀이 완료·카드화·Anki 싱크). **5.2 「행렬식의 계산과 응용」 정리 완료**: 8단계 구조(고윳값 λ 수위조절·식6 상세·detAB 증명 3수 구조·EX1/EX2 상세·큰 공식 재구성), 하이라이트 31개(2차 5개 포함) 연결, 추출 이미지 16개 + 자작 SVG 1개(치환 6개 그리드), OCR 교정 15건, 연습문제 8제(1·2·3·5·7·10·13·16) 해설, 스스로 점검 8문 전부 사용자 #card 카드화·Anki 싱크 완료. 5.2는 굿노트 없이 진행(사용자 판단, 필기 적으면 생략 가능 선례). **현재 단계: 사용자가 5.2 연습문제 풀 예정** (질문에 "수정하지 말고 답변만"이 붙으면 파일을 건드리지 않는다). 5장 쪽번호 오프셋: 책 쪽 = PDF 쪽 + 278 (총 24쪽, 5.2 본문은 PDF 11~16·연습문제 1~19번은 PDF 16~19쪽). 해답집 Problem Set 5.1은 해답 PDF 83~84쪽, 5.2는 84~86쪽. ⚠️ 해답집 불일치 이력 (한국어판 행렬 원문부터 재확인할 것): 5.1 문제 6 (한국어판 정답 −1, 해답집 4), 5.2 문제 2 detC (한국어판 [A 0;0 A] 기준 4, 해답집 −1), 그리고 **해답집 5.2의 13·14·15·17·18번은 한국어판과 아예 다른 문제** (원서 구성 차이, 시계 벡터 등) → 이 번호들은 해답집을 무시하고 직접 풀이·검산. 다음: 5.3 (마지막 절, 「5장 고찰」+「이 단원 핵심문제」 동반). 2장에서 확립된 방식(🧩 행렬 형태 명시·숫자 예제 먼저·단계 지도·흔한 오해 교정·자작 SVG) 계속 적용. 이해 안 되는 부분 요청 시 해당 블록만 보강하는 패턴 유지. - ## 단원 PDF 내보내기 (아이디어, 요청 시 구현) - 사용자가 "n단원 pdf로 만들어줘"라고 하면, 그 단원 페이지 + 거기 링크된 소단원 페이지들을 묶어 Logseq에서 보이는 느낌의 PDF 한 부로 만든다 (2026-08-20 가능성만 확인해 둔 상태, **아직 미구현**. 사용자가 필요할 때 요청하기로 함). - 파이프라인: ① 단원 페이지에서 `[[x.y ...]]` 링크를 수집해 페이지 목록 구성 → ② Logseq 문법 전처리: 속성 줄(`id::`·`deck::`·`collapsed::` 등) 숨김, `{{c1 ...}}` cloze는 괄호 벗겨 일반 텍스트화, `((uuid))`는 hls 페이지에서 원문 텍스트를 찾아 "📌 ... 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{:bounding {:x1 356.9878234863281, + :y1 357.92100524902344, + :x2 506.4344482421875, + :y2 371.1154327392578, + :width 716, + :height 1012.6281590388325}, + :rects ({:x1 356.9878234863281, + :y1 357.92100524902344, + :x2 506.4344482421875, + :y2 371.1154327392578, + :width 716, + :height 1012.6281590388325}), + :page 11}, + :content {:text "가역 행렬 A는 detA≠ 0 "}, + :properties {:color "yellow"}} + {:id #uuid "6a943a00-fc7a-40ae-8856-7a6371a40c25", + :page 11, + :position {:bounding {:x1 63.498260498046875, + :y1 357.92100524902344, + :x2 602.1060791015625, + :y2 398.6544952392578, + :width 716, + :height 1012.6281590388325}, + :rects ({:x1 578.3745727539062, + :y1 357.92100524902344, + :x2 602.1060791015625, + :y2 371.1154327392578, + :width 716, + :height 1012.6281590388325} + {:x1 63.498260498046875, + :y1 384.76560974121094, + :x2 183.3550262451172, + :y2 398.6544952392578, + :width 716, + :height 1012.6281590388325}), + :page 11}, + :content {:text "특이행렬 A는 detA =0"}, + :properties {:color "yellow"}} + {:id #uuid "6a943a60-ddf1-47f7-93fb-85ab1f858a28", + :page 11, + :position {:bounding {:x1 63.498260498046875, + :y1 384.76560974121094, + :x2 601.0880737304688, + :y2 425.5099639892578, + :width 716, + :height 1012.6281590388325}, + :rects ({:x1 270.99609375, + :y1 384.76560974121094, + :x2 601.0880737304688, + :y2 398.6544952392578, + :width 716, + :height 1012.6281590388325} + {:x1 63.498260498046875, + :y1 412.31553649902344, + :x2 505.50128173828125, + :y2 425.5099639892578, + :width 716, + :height 1012.6281590388325}), + :page 11}, + :content {:text "Aziz에서 고 윳값 와 고유 벡터 는 고 윳값 방정식( AI)z=0 을 만족한다 . 즉 A- J가 특이행렬 이고 det(AI ) = 0 "}, + :properties {:color "yellow"}} + {:id #uuid "6a943a8d-513d-4a41-842f-dc51c4309ade", + :page 11, + :position {:bounding {:x1 62.499996185302734, + :y1 504.39451599121094, + :x2 598.7630004882812, + :y2 543.2183074951172, + :width 716, + :height 1012.6281590388325}, + :rects ({:x1 418.2834167480469, + :y1 504.39451599121094, + :x2 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"6a943aa6-c986-4f50-b6ae-fa2af53fff53", + :page 11, + :position {:bounding {:x1 64, + :y1 721.2216796875, + :x2 604, + :y2 761.2216796875, + :width 716, + :height 1012.6281590388325}, + :rects (), + :page 11}, + :content {:text "[:span]", :image 1788099238051}, + :properties {:color "yellow"}} + {:id #uuid "6a943ace-dea6-4c62-bf04-eb09243c01ef", + :page 11, + :position {:bounding {:x1 208.7781982421875, + :y1 896.9292449951172, + :x2 446.9326171875, + :y2 908.7347869873047, + :width 716, + :height 1012.6281590388325}, + :rects ({:x1 208.7781982421875, + :y1 896.9292449951172, + :x2 446.9326171875, + :y2 908.7347869873047, + :width 716, + :height 1012.6281590388325}), + :page 11}, + :content {:text "직교행렬 Q 의 행렬식은 1 또는 -1 이다."}, + :properties {:color "yellow"}} + {:id #uuid "6a943b00-83df-4cd4-b047-1f23ac2d8d6a", + :page 12, + :position {:bounding {:x1 105, + :y1 141.9716796875, + :x2 650, + :y2 181.9716796875, + :width 716, + :height 1012.6281590388325}, + :rects (), + :page 12}, + :content {:text "[:span]", :image 1788099328536}, + :properties {:color "yellow"}} + {:id #uuid "6a943b6f-c65a-4e0b-b6e8-afae671d15f7", + :page 12, + :position {:bounding {:x1 106.82508087158203, + :y1 206.7816619873047, + :x2 621.4225463867188, + :y2 220.67054748535156, + :width 716, + :height 1012.6281590388325}, + :rects ({:x1 106.82508087158203, + :y1 206.7816619873047, + :x2 621.4225463867188, + :y2 220.67054748535156, + :width 716, + :height 1012.6281590388325}), + :page 12}, + :content {:text "A = LU 이면 detA= ( detZ ) (detU )=detU 이다 . 삼각행렬 U 의 행렬식은 피벗들의 곱"}, + :properties {:color "yellow"}} + {:id #uuid "6a943b74-de8b-439f-80f0-6a3e074bc78b", + :page 12, + :position {:bounding {:x1 106.82508087158203, + :y1 234.3315887451172, + :x2 509.8168029785156, + :y2 247.52601623535156, + :width 716, + :height 1012.6281590388325}, + :rects ({:x1 106.82508087158203, + :y1 234.3315887451172, + :x2 509.8168029785156, + :y2 247.52601623535156, + :width 716, + :height 1012.6281590388325}), + :page 12}, + :content {:text "PA = LU이면 행 교환이 존재하므로 det=1 또는 -1 이다(치환행렬"}, + :properties {:color "yellow"}} + {:id #uuid "6a943b90-0266-4228-baf4-227d765daec4", + :page 12, + :position {:bounding {:x1 102, + :y1 474.8330078125, + :x2 656, + :y2 683.8330078125, + :width 716, + :height 1012.6281590388325}, + :rects (), + :page 12}, + :content {:text "[:span]", :image 1788099472392}, + :properties {:color "yellow"}} + {:id #uuid "6a943be0-7639-45fc-ae5e-fa811c704e4f", + :page 12, + :position {:bounding {:x1 109.75476837158203, + :y1 796.3433074951172, + :x2 651.5841674804688, + :y2 837.0876312255859, + :width 716, + :height 1012.6281590388325}, + :rects ({:x1 401.5190734863281, + :y1 796.3433074951172, + :x2 651.5841674804688, + :y2 810.2321624755859, + :width 716, + :height 1012.6281590388325} + {:x1 109.75476837158203, + :y1 823.8931732177734, + :x2 141.44004821777344, + :y2 837.0876312255859, + :width 716, + :height 1012.6281590388325}), + :page 12}, + :content {:text "행렬 A 의 두 행이 동일하면 A의 행렬식 은0이다"}, + :properties {:color "yellow"}} + {:id #uuid "6a943c8c-83fb-4053-afa3-f1a2b63d5179", + :page 13, + :position {:bounding {:x1 215.41574096679688, + :y1 71.34332275390625, + :x2 574.5114135742188, + :y2 85.9266357421875, + :width 716, + :height 1012.6281590388325}, + :rects ({:x1 215.41574096679688, + :y1 71.34332275390625, + :x2 574.5114135742188, + :y2 85.9266357421875, + :width 716, + :height 1012.6281590388325}), + :page 13}, + :content {:text "j행에서 행 에 d를 곱한 것을 빼도 detA는 변하지 않는다"}, + :properties {:color "yellow"}} + {:id #uuid "6a943c91-84b2-4cc8-86de-b1018447eac9", + :page 13, + :position {:bounding {:x1 140, + :y1 90.77734375, + :x2 615, + :y2 167.77734375, + :width 716, + :height 1012.6281590388325}, + :rects (), + :page 13}, + :content {:text "[:span]", :image 1788099729531}, + :properties {:color "yellow"}} + {:id #uuid "6a943cf9-6d55-4024-ba3b-d3a3996e7755", + :page 13, + :position {:bounding {:x1 61.425777435302734, + :y1 201.81207275390625, + :x2 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(), + :page 13}, + :content {:text "[:span]", :image 1788099890542}, + :properties {:color "yellow"}} + {:id #uuid "6a943dc0-8127-417c-9142-7f364dec2cbc", + :page 13, + :position {:bounding {:x1 61.425777435302734, + :y1 692.3068161010742, + :x2 607.1199951171875, + :y2 764.7460861206055, + :width 716, + :height 1012.6281590388325}, + :rects ({:x1 433.0946044921875, + :y1 692.3068161010742, + :x2 607.1199951171875, + :y2 705.5012741088867, + :width 716, + :height 1012.6281590388325} + {:x1 62.424041748046875, + :y1 721.9834518432617, + :x2 604.108642578125, + :y2 735.1779098510742, + :width 716, + :height 1012.6281590388325} + {:x1 61.425777435302734, + :y1 750.8571701049805, + :x2 442.0025634765625, + :y2 764.7460861206055, + :width 716, + :height 1012.6281590388325}), + :page 13}, + :content {:text "항등행렬 / 1 열을 로 바꾼다 . 이렇게 바꾼 삼각 행렬 M 의 행렬식 은 이다. M 에 행렬 A 를 곱하면 1열 은 Ar 가 되고,가정에 따라 Ax= b이다 . B. 의 나머지 열 은 행렬 A 와 동일 하다."}, + :properties {:color "yellow"}} + {:id #uuid "6a943dc5-2624-4f08-b3e4-b2f6a1d99dc2", + :page 13, + :position {:bounding {:x1 130, + :y1 767.3330078125, + :x2 607, + :y2 847.3330078125, + :width 716, + :height 1012.6281590388325}, + :rects (), + :page 13}, + :content {:text "[:span]", :image 1788100037189}, + :properties {:color "yellow"}} + {:id #uuid "6a943dfb-3873-4e19-a53f-6bd6e3d16dc2", + :page 14, + :position {:bounding {:x1 144, + :y1 68.6943359375, + :x2 652, + :y2 130.6943359375, + :width 716, + :height 1012.6281590388325}, + :rects (), + :page 14}, + :content {:text "[:span]", :image 1788100091108}, + :properties {:color "yellow"}} + {:id #uuid "6a943e05-9a21-41e1-b292-8eb0ff3c1eb4", + :page 14, + :position {:bounding {:x1 146, + :y1 216.6943359375, + :x2 645, + :y2 288.6943359375, + :width 716, + :height 1012.6281590388325}, + :rects (), + :page 14}, + :content {:text "[:span]", :image 1788100100768}, + :properties {:color "yellow"}} + {:id #uuid "6a943e84-a662-40ad-9401-d69a0546bb42", + :page 14, + :position {:bounding {:x1 97, + :y1 596.5546875, + :x2 659, + :y2 755.5546875, + :width 716, + :height 1012.6281590388325}, + :rects (), + :page 14}, + :content {:text "[:span]", :image 1788100227948}, + :properties {:color "yellow"}} + {:id #uuid "6a943ef3-97cd-4c00-9e44-307fc49d3168", + :page 15, + :position {:bounding {:x1 60, + :y1 755.138671875, + :x2 586, + :y2 820.138671875, + :width 716, + :height 1012.6281590388325}, + :rects (), + :page 15}, + :content {:text "[:span]", :image 1788100339170}, + :properties {:color "yellow"}} + {:id #uuid "6a943efb-271f-4733-ab93-21fa9bd6557c", + :page 15, + :position {:bounding {:x1 194.74826049804688, + :y1 489.8111877441406, + :x2 552.888916015625, + :y2 504.39453125, + :width 716, + :height 1012.6281590388325}, + :rects ({:x1 194.74826049804688, + :y1 489.8111877441406, + :x2 552.888916015625, + :y2 504.39453125, + :width 716, + :height 1012.6281590388325}), + :page 15}, + :content {:text "크레이머 규칙 에서 각 B 의 행렬식은 행렬 A의 여인이다."}, + :properties {:color "yellow"}} + {:id #uuid "6a943f06-f385-4965-81c1-6706552d7c05", + :page 15, + :position {:bounding {:x1 65.22351837158203, + :y1 462.4565734863281, + :x2 303.26605224609375, + :y2 479.1232604980469, + :width 716, + :height 1012.6281590388325}, + :rects ({:x1 65.22351837158203, + :y1 462.4565734863281, + :x2 303.26605224609375, + :y2 479.1232604980469, + :width 716, + :height 1012.6281590388325}), + :page 15}, + :content {:text "A- '는 행렬 A의 여인 들을 포함한다 "}, + :properties {:color "yellow"}} + {:id #uuid "6a94462d-4af3-413d-b22b-6911e58b9b2c", + :page 13, + :position {:bounding {:x1 61.425777435302734, + :y1 336.74046325683594, + :x2 602.18701171875, + :y2 389.07334899902344, + :width 716, + :height 1012.6281590388325}, + :rects ({:x1 554.7417602539062, + :y1 336.74046325683594, + :x2 602.18701171875, + :y2 347.15708923339844, + :width 716, + :height 1012.6281590388325} + {:x1 495.388427734375, + :y1 342.9253387451172, + :x2 553.8519897460938, + :y2 360.2864227294922, + :width 716, + :height 1012.6281590388325} + {:x1 560.622802734375, + :y1 356.2933807373047, + :x2 596.2056884765625, + :y2 366.01560974121094, + :width 716, + :height 1012.6281590388325} + {:x1 61.425777435302734, + :y1 372.4066619873047, + :x2 392.5886535644531, + :y2 389.07334899902344, + :width 716, + :height 1012.6281590388325}), + :page 13}, + :content {:text "( A)= detAB det B가 앞서 살펴본 성질 2. 2. 2 을 만족함을 보이고자 한다"}, + :properties {:color "yellow"}} + {:id #uuid "6a94464a-ed55-45c7-96d4-135fa44a9387", + :page 14, + :position {:bounding {:x1 85, + :y1 340.6103515625, + :x2 662, + :y2 578.6103515625, + :width 716, + :height 1012.6281590388325}, + :rects (), + :page 14}, + :content {:text "[:span]", :image 1788102217715}, + :properties {:color "yellow"}} + {:id #uuid "6a944666-5084-47c1-b9a8-0960750738ea", + :page 15, + :position {:bounding {:x1 65.22351837158203, + :y1 348.4917507171631, + :x2 546.0828857421875, + 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zB5e)^?z{}b@L~tYrU&X21ijezQpM_DN65;4zL2ig zCPQDxwWTLw#(R%02R;PVC1A-D9%o{yW0d{2C6*84N^}VQA+iO`oMU~VscJQBtB{Gp zP^7B8``0ZMc#R^B4SNnHDjq=-%*n?L5+doTdyRxdvn$!kYx7~xa^4q})0pb#T#Su+ z`r?VPXEkf 3$ 이면 매우 느린 방법) + - 출발점은 5.1의 핵심 판정이다. 행렬식 하나로 가역/특이가 갈린다: + - 📌 원문 가역행렬 판정 (p.289): ((6a9439f3-8c77-4ce4-bf40-de233cd12495)) + - 📌 원문 특이행렬 판정 (p.289): ((6a943a00-fc7a-40ae-8856-7a6371a40c25)) + - 💡 선행 개념 수위 조절: 첫 문단에 설명 없이 등장하는 고윳값(eigenvalue) $\lambda$ 는 6장의 주인공이다. 지금은 다음 흐름만 받아들이면 충분하다. + - $Ax = \lambda x$ 를 만족하는 특별한 수 $\lambda$ 를 찾고 싶다. 식을 옮기면 $(A - \lambda I)x = 0$ 이고, $x \neq 0$ 인 해가 필요하다. + - 어떤 행렬에 대해 $x \neq 0$ 인 해가 존재하려면 그 행렬이 특이행렬이어야 했다 (3장의 영공간 이야기). 그러므로 조건은 "$A - \lambda I$ 가 특이행렬"이다. + - "특이행렬인가?"를 수 하나로 판정하는 도구가 행렬식이다. 그래서 $\lambda$ 에 관한 방정식 $\det(A - \lambda I) = 0$ 이 등장한다. + - 즉 이 문단은 "행렬식이 6장의 핵심 도구가 된다"는 예고편이다. 지금 $\lambda$ 를 구하는 방법까지 알 필요는 없다 (6장에서 제대로 배운다). + - 📌 원문 고윳값 방정식 (p.289): ((6a943a60-ddf1-47f7-93fb-85ab1f858a28)) + - #### [2단계] 유용한 성질 4가지 + - **성질 (1): 삼각행렬과 대각행렬은 주대각성분만 곱하면 끝** + - 📌 원문 (p.289): ((6a943a8d-513d-4a41-842f-dc51c4309ade)) + - 📌 원문 식 (1) (p.289): [🔗 PDF 위치](((6a943a93-ffe5-493c-beec-c2e373dfec63))) + - ![삼각행렬·대각행렬의 행렬식, 식 1 (p.289)](../assets/5장_p289_삼각대각행렬_식1.png) + - 🔢 이유를 상삼각행렬로 확인: 1열로 여인수 전개하면 1열에 $a$ 하나뿐이라 항이 하나만 살아남고, 남은 행렬도 다시 삼각행렬이라 같은 일이 반복된다. + - $\det\begin{bmatrix} a & b & c \\ 0 & q & r \\ 0 & 0 & z \end{bmatrix} = a \cdot \det\begin{bmatrix} q & r \\ 0 & z \end{bmatrix} = a \cdot qz$ + - **성질 (2): 전치해도 행렬식은 그대로** + - 📌 원문 식 (2) (p.289): [🔗 PDF 위치](((6a943a9f-5ef4-4a1a-aeae-6b7556fdeb58))) + - ![전치행렬의 행렬식, 식 2 (p.289)](../assets/5장_p289_전치행렬_식2.png) + - 💡 이 성질 덕분에 "행에 대한 규칙"이 전부 "열에 대한 규칙"으로도 성립한다 (행 교환 ↔ 열 교환, 행 선형성 ↔ 열 선형성). + - 5.1에서 "두 열을 교환해도 부호가 바뀐다"고 괄호로 덧붙였던 근거가 바로 이것이다. + - **성질 (3): 곱의 행렬식은 행렬식의 곱** (증명은 5단계에서) + - 📌 원문 식 (3) (p.289): [🔗 PDF 위치](((6a943aa6-c986-4f50-b6ae-fa2af53fff53))) + - ![두 행렬의 곱, 식 3 (p.289)](../assets/5장_p289_두행렬의곱_식3.png) + - ⚠️ 합에는 이런 규칙이 없다: $\det(A + B) \neq \det A + \det B$. + - 반례: $I + I = 2I$ 의 행렬식은 $2$ 가 아니라 $2^n$ 이다 (행 $n$ 개가 각각 2배). + - **직교행렬: 행렬식은 1 또는 −1** + - 📌 원문 (p.289): ((6a943ace-dea6-4c62-bf04-eb09243c01ef)) + - 유도: $Q^{\mathrm{T}}Q = I$ 에 성질 (2)와 (3)을 적용한다. + - $(\det Q^{\mathrm{T}})(\det Q) = \det I = 1$ + - $\det Q^{\mathrm{T}} = \det Q$ 이므로 $(\det Q)^2 = 1$, 즉 $\det Q = \pm 1$. + - 💡 기하 감각: 직교행렬은 회전·반사처럼 길이를 보존하는 변환이라 "부피 배율"이 1일 수밖에 없다 (행렬식 = 부피 배율 해석은 5.3에서). + - **성질 (4): 행렬식 = ±(피벗들의 곱)** (실전 계산법) + - 📌 원문 식 (4) (p.290): [🔗 PDF 위치](((6a943b00-83df-4cd4-b047-1f23ac2d8d6a))) + - ![가역행렬의 행렬식은 ±피벗들의 곱, 식 4 (p.290)](../assets/5장_p290_피벗들의곱_식4.png) + - 📌 원문 LU와 행렬식 (p.290): ((6a943b6f-c65a-4e0b-b6e8-afae671d15f7)) + - 📌 원문 PA = LU와 부호 (p.290): ((6a943b74-de8b-439f-80f0-6a3e074bc78b)) + - $A = LU$ 이면: $\det A = (\det L)(\det U) = \det U = U_{11}U_{22} \cdots U_{nn}$. + - $L$ 은 주대각이 전부 1인 삼각행렬이라 $\det L = 1$, $U$ 는 삼각행렬이라 피벗들의 곱 (성질 1). + - 행 교환이 필요했다면 $PA = LU$ 이고 $\det P = \pm 1$ 이므로, $\det A = \pm(\text{피벗들의 곱})$ 이 된다. + - 💡 소거법이 선형대수학을 위한 거의 모든 컴퓨터 시스템의 행렬식 계산법이다. 비용은 곱셈 약 $n^3/3$ 번 (2장의 소거 비용 그대로). + - "왜 소거가 행렬식을 바꾸지 않는지"는 4단계에서 증명한다. 그 증명이 끝나야 이 공식이 완성된다. + - #### [3단계] 모든 성질의 뿌리: 기본 성질 ❶❷❸ + - 지금까지 나온 성질들은 전부 다음 3개에서 유도된다. 이 3개가 행렬식의 "공리"다: + - ❶ 항등행렬: $\det I = 1$ + - ❷ 행 교환: 두 행을 교환하면 $\det A$ 는 $-\det A$ 로 부호가 바뀐다 + - ❸ 선형성: 1행이 $cv + dw$ 이면 행렬식도 두 행렬식의 선형결합으로 쪼개진다 (식 5). 나머지 행은 그대로 둔 채 1행에 대해서만. + - 📌 원문 기본 성질 ❶❷❸과 식 (5) (p.290): [🔗 PDF 위치](((6a943b90-0266-4228-baf4-227d765daec4))) + - ![행렬식의 기본 성질 3가지와 식 5 (p.290)](../assets/5장_p290_기본성질3가지_식5.png) + - 💡 왜 "1행"만 말하나: 성질 ❷로 아무 행이나 1행 자리로 데려올 수 있으므로 사실상 모든 행에 대해 선형이다 (책의 표현: "1행이 특별한 것은 아니다"). + - 💡 유일성 (5단계에서 쓸 핵심 무기): 성질 ❶❷❸만 있으면 어떤 행렬이든 행렬식 값이 완전히 결정된다. + - 이유: 행 빼기(❸에서 유도, 4단계)와 행 교환(❷)만으로 임의의 행렬을 삼각행렬로 만들 수 있고, 삼각행렬의 값은 ❶과 ❸이 결정하기 때문. + - 따라서 ❶❷❸을 모두 만족하는 함수는 행렬식 딱 하나뿐이다. 어떤 함수가 이 3개를 만족함을 보이면 그 함수는 행렬식이다. + - ❷에서 바로 나오는 사실: **두 행이 동일하면 행렬식은 0**. + - 📌 원문 (p.290): ((6a943be0-7639-45fc-ae5e-fa811c704e4f)) + - 같은 두 행을 교환하면 행렬은 그대로인데 부호는 바뀌어야 하므로 $\det A = -\det A$, 즉 $\det A = 0$. + - 5.1에서 $AC^{\mathrm{T}}$ 의 비대각성분을 0으로 만든 바로 그 규칙이다. + - ❸에서 바로 나오는 사실: 한 행에 $c$ 를 곱하면 행렬식도 $c$ 배가 된다. + - 행렬 전체에 곱하면: $\det(cA) = c^n \det A$ (행이 $n$ 개니까 $c$ 배가 $n$ 번). + - #### [4단계] 소거는 행렬식을 바꾸지 않는다 (식 6·7) + - 🔢 먼저 숫자로 확인: + - $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ 의 행렬식은 $4 - 6 = -2$. + - 소거 한 걸음: 2행에서 1행의 3배를 빼면 $\begin{bmatrix} 1 & 2 \\ 0 & -2 \end{bmatrix}$, 행렬식은 $1 \times (-2) = -2$. 그대로다. + - "빼기를 했는데 왜 값이 안 변할까"라는 의문이 자연스럽다. 선형성 ❸으로 쪼개면 답이 보인다: + - 소거 후의 2행은 $(3, 4) - 3 \cdot (1, 2)$ 라는 선형결합이다. 2행에 대한 선형성으로 두 행렬식으로 분리하면: + - $$\det\begin{bmatrix} 1 & 2 \\ 3 - 3 \cdot 1 & 4 - 3 \cdot 2 \end{bmatrix} = \det\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} - 3 \cdot \det\begin{bmatrix} 1 & 2 \\ 1 & 2 \end{bmatrix}$$ + - 뒤 행렬식은 1행과 2행이 같으므로 0 (3단계의 "같은 행 2개" 규칙). + - 남는 것은 $\det A - 3 \cdot 0 = \det A$. 이것이 전부다. + - 일반형이 식 (6)이다: $j$ 행에서 $i$ 행에 $d$ 를 곱한 것을 빼도 $\det A$ 는 변하지 않는다. + - 📌 원문 (p.291): ((6a943c8c-83fb-4053-afa3-f1a2b63d5179)) + - 📌 원문 식 (6) (p.291): [🔗 PDF 위치](((6a943c91-84b2-4cc8-86de-b1018447eac9))) + - ![행 빼기는 행렬식을 바꾸지 않는다, 식 6 (p.291)](../assets/5장_p291_행빼기불변_식6.png) + - 말로 풀면: 빼낸 부분은 "$j$ 행 자리에 $i$ 행이 한 번 더 들어간 행렬"의 행렬식 $d$ 배로 떨어져 나가는데, 그 행렬은 같은 행이 2개라서 행렬식이 0이다. + - ⚠️ 흔한 오해 교정: "한 행에 $d$ 를 곱하면 행렬식도 $d$ 배가 된다면서(❸), 왜 $d$ 배를 빼는 건 괜찮은가?"라고 생각하기 쉽지만 둘은 다른 연산이다. + - $j$ 행 자체에 $d$ 를 곱하면: 행렬식도 $d$ 배가 된다. + - $j$ 행에서 **다른 행**($i$ 행)의 $d$ 배를 빼면: 그 부분이 "같은 행 2개" 행렬식으로 분리되어 0으로 사라진다. 그래서 불변. + - 결론: (행 교환이 없는) 소거 과정 전체가 행렬식을 보존한다. 그래서 식 (7), 즉 성질 (4)가 증명된다. + - 📌 원문 (p.291): ((6a943cf9-6d55-4024-ba3b-d3a3996e7755)) + - 📌 원문 식 (7) (p.291): [🔗 PDF 위치](((6a943d04-373b-4865-b6d7-26b3b6f77324))) + - ![행 교환이 없을 때 detA = 피벗들의 곱, 식 7 (p.291)](../assets/5장_p291_행교환없을때_피벗곱_식7.png) + - $A \to U$ 로 가는 동안 행렬식은 한 번도 변하지 않았고, 도착한 $U$ 는 삼각행렬이라 피벗들의 곱이다. + - #### [5단계] det(AB) = (detA)(detB) 증명 + - 증명 전략부터: 양변을 각각 전개해서 비교하는 것이 아니다. 3단계의 유일성, 즉 "성질 ❶❷❸을 모두 만족하는 함수는 행렬식뿐"을 무기로 쓴다. + - [1수] $B$ 를 고정하고 ($\det B \neq 0$ 가정), $A$ 에 대한 함수를 하나 만든다: + - $$D(A) = \frac{\det AB}{\det B}$$ + - [2수] 이 $D$ 가 성질 ❶❷❸을 전부 만족함을 확인한다. + - [3수] 유일성에 의해 $D(A) = \det A$. 분모를 넘기면 $\det AB = (\det A)(\det B)$. 끝. + - 📌 원문 증명 전략 (p.291): ((6a94462d-4af3-413d-b22b-6911e58b9b2c)) + - [2수]의 확인을 하나씩 뜯어보면: + - ❶ 확인: $A = I$ 이면 $AB = B$ 이므로 $D(I) = \det B / \det B = 1$. 통과. + - ❷ 확인: $A$ 의 두 행을 교환하면 $AB$ 의 같은 두 행이 교환된다. + - 왜냐하면 $AB$ 의 $i$ 행은 ($A$ 의 $i$ 행) $\times B$ 이기 때문이다 (행렬 곱은 행 단위로 작동, 2장). $A$ 의 행 순서를 바꾸면 결과의 행 순서도 똑같이 바뀐다. + - $AB$ 의 두 행이 교환되면 $\det AB$ 의 부호가 바뀌고, 분모 $\det B$ 는 그대로이므로 $D(A)$ 의 부호도 바뀐다. 통과. + - ❸ 확인: $A$ 의 1행에 $c$ 를 곱하면 $AB$ 의 1행도 $c$ 배가 된다. $A$ 의 1행이 $v + w$ 이면 $AB$ 의 1행은 $vB + wB$ 다. + - 그러면 식 (5)가 $\det(AB)$ 를 두 행렬식으로 분해해 주므로 $D$ 도 1행에 대해 선형이다. 통과. + - 📌 원문 ❶❷❸ 확인 (p.291): [🔗 PDF 위치](((6a943d32-0ad3-46e3-bc58-5936bbb3736c))) + - ![D(A)가 세 성질을 만족한다 (p.291)](../assets/5장_p291_detAB증명_성질확인.png) + - 🔢 숫자 검증: $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ ($\det = -2$), $B = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$ ($\det = 6$). + - $AB = \begin{bmatrix} 2 & 6 \\ 6 & 12 \end{bmatrix}$, $\det AB = 24 - 36 = -12 = (-2)(6)$. 일치. + - 남은 구멍 하나: $\det B = 0$ 인 경우는? + - $B$ 가 특이행렬이면 $Bx = 0$ 인 $x \neq 0$ 이 있고, 그러면 $(AB)x = A(Bx) = 0$ 이므로 $AB$ 도 특이행렬이다. + - 따라서 양변 모두 0: $\det AB = 0 = (\det A) \cdot 0$. 곱 규칙은 이 경우에도 성립한다. + - 💡 이 증명 패턴("후보 함수가 ❶❷❸을 만족함을 확인하면 그 함수는 행렬식이다")은 행렬식 증명의 표준 무기다. 지루한 대수 전개 없이 성질만으로 끝난다. + - #### [6단계] 크레이머 규칙 + - 목표: $Ax = b$ 의 해의 각 성분 $x_j$ 를 행렬식들의 비율로 쓰기. + - 핵심 아이디어는 "정체를 아는 행렬 곱" 하나다. 항등행렬 $I$ 의 1열을 $x$ 로 바꾼 행렬을 $M_1$ 이라 하자. + - 📌 원문 (p.291): ((6a943dc0-8127-417c-9142-7f364dec2cbc)) + - 🧩 형태 ($3 \times 3$ 기준): + - $$A M_1 = A \begin{bmatrix} x_1 & 0 & 0 \\ x_2 & 1 & 0 \\ x_3 & 0 & 1 \end{bmatrix} = \begin{bmatrix} b_1 & a_{12} & a_{13} \\ b_2 & a_{22} & a_{23} \\ b_3 & a_{32} & a_{33} \end{bmatrix} = B_1$$ + - 왜 이렇게 되나 (열별로 확인): $AM_1$ 의 1열은 $A x = b$ (가정). 2열은 $A \cdot (0, 1, 0) = A$ 의 2열. 3열도 $A$ 의 3열 그대로. + - $M_1$ 의 행렬식: 하삼각행렬이므로 대각 곱, 즉 $\det M_1 = x_1 \cdot 1 \cdot 1 = x_1$. + - 📌 원문 식 (8) (p.291): [🔗 PDF 위치](((6a943dc5-2624-4f08-b3e4-b2f6a1d99dc2))) + - ![핵심 아이디어 AM₁ = B₁, 식 8 (p.291)](../assets/5장_p291_크레이머_핵심아이디어_식8.png) + - 식 (8)의 양변에 곱 규칙(5단계)을 적용하면 $x_1$ 이 튀어나온다: + - $(\det A)(\det M_1) = \det B_1$ 이고 $\det M_1 = x_1$ 이므로 $x_1 = \det B_1 / \det A$. + - 📌 원문 식 (9) (p.292): [🔗 PDF 위치](((6a943dfb-3873-4e19-a53f-6bd6e3d16dc2))) + - ![곱셈 규칙, 식 9 (p.292)](../assets/5장_p292_곱셈규칙_식9.png) + - $x_2$ 도 같은 아이디어: $I$ 의 2열을 $x$ 로 바꾼 $M_2$ 를 쓰면 $AM_2 = B_2$ ($A$ 의 2열이 $b$ 로 바뀐 행렬), $\det M_2 = x_2$. + - 📌 원문 식 (10) (p.292): [🔗 PDF 위치](((6a943e05-9a21-41e1-b292-8eb0ff3c1eb4))) + - ![동일한 아이디어 AM₂ = B₂, 식 10 (p.292)](../assets/5장_p292_동일한아이디어_식10.png) + - [정리] 크레이머 규칙 (식 11): $\det A \neq 0$ 이면 $x_j = \det B_j / \det A$. + - $B_j$ 는 $A$ 의 $j$ 열이 벡터 $b$ 로 대치된 행렬이다. + - 📌 원문 정리 (p.292): [🔗 PDF 위치](((6a943e84-a662-40ad-9401-d69a0546bb42))) + - ![크레이머 규칙 정리, 식 11 (p.292)](../assets/5장_p292_크레이머규칙_정리_식11.png) + - 🔢 예제 EX 1 자세히 (p.292): $3x_1 + 4x_2 = 2$, $5x_1 + 6x_2 = 4$ + - 📌 원문 EX 1 (p.292): [🔗 PDF 위치](((6a94464a-ed55-45c7-96d4-135fa44a9387))) + - ![EX 1 크레이머 규칙 계산 (p.292)](../assets/5장_p292_EX1_크레이머_예제.png) + - [1] 행렬식 3개 준비: + - $\det A = \begin{vmatrix} 3 & 4 \\ 5 & 6 \end{vmatrix} = 18 - 20 = -2$ + - $\det B_1 = \begin{vmatrix} 2 & 4 \\ 4 & 6 \end{vmatrix} = 12 - 16 = -4$ (1열을 $b = (2, 4)$ 로 교체) + - $\det B_2 = \begin{vmatrix} 3 & 2 \\ 5 & 4 \end{vmatrix} = 12 - 10 = 2$ (2열을 $b$ 로 교체) + - [2] 나누기: $x_1 = -4/(-2) = 2$, $x_2 = 2/(-2) = -1$. + - [3] 검산 (직접 대입): $3 \cdot 2 + 4 \cdot (-1) = 2$, $5 \cdot 2 + 6 \cdot (-1) = 4$. 둘 다 맞다. + - ⚠️ 비용: 크레이머 규칙은 아름답지만 느리다. + - $(n+1)$ 개의 행렬식이 필요하고, 큰 공식 기준으로 각각 $n!$ 개 항이라 전체 $(n+1)!$ 개 항이다 (책의 표현: "방정식을 이런 식으로 푸는 것은 터무니없다"). + - 실전 계산은 소거 ($n^3/3$). 크레이머 규칙의 진짜 용도는 EX 2처럼 문자가 든 계산과 "해를 공식으로 써야 하는" 이론 전개다. + - #### [7단계] A⁻¹의 공식: 여인수로 완성 + - 5.1에서 만든 $A^{-1} = C^{\mathrm{T}} / \det A$ 가 크레이머 규칙에서 다시 유도된다. + - 🔢 예제 EX 2 자세히 ($n = 2$, p.293): $AA^{-1} = I$ 를 열별로 읽으면 연립방정식 2개가 된다. + - $A^{-1}$ 의 두 열을 $x$, $y$ 라 하면: $Ax = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$, $Ay = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$. + - $x$ 에 크레이머 규칙 ($A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$): + - $x_1 = \begin{vmatrix} 1 & b \\ 0 & d \end{vmatrix} / |A| = d / |A|$ + - $x_2 = \begin{vmatrix} a & 1 \\ c & 0 \end{vmatrix} / |A| = -c / |A|$ + - $y$ 에 크레이머 규칙: + - $y_1 = \begin{vmatrix} 0 & b \\ 1 & d \end{vmatrix} / |A| = -b / |A|$ + - $y_2 = \begin{vmatrix} a & 0 \\ c & 1 \end{vmatrix} / |A| = a / |A|$ + - 등장한 4개의 값 $d, -c, -b, a$ 는 전부 $A$ 의 여인수다. 5.1의 여인수행렬 $C = \begin{bmatrix} d & -c \\ -b & a \end{bmatrix}$ 그대로: + - 📌 원문 (p.293): ((6a944666-5084-47c1-b9a8-0960750738ea)) + - $$A^{-1} = \frac{1}{ad - bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix} = \frac{C^{\mathrm{T}}}{\det A}$$ + - 왜 여인수가 나오나: $b$ 가 $I$ 의 열(1 하나, 나머지 0)이면 $\det B_j$ 를 그 1이 있는 열로 여인수 전개할 때 항이 딱 하나만 살아남기 때문이다. + - 📌 원문 (p.293): ((6a943efb-271f-4733-ab93-21fa9bd6557c)) + - $n = 3$ 확인: $Ax = (1, 0, 0)$ 이면 $\det B_1 = C_{11}$, $\det B_2 = -(a_{21}a_{33} - a_{23}a_{31}) = C_{12}$. 필요한 음수 부호까지 여인수 정의에 자동으로 들어 있다. + - 일반 공식 (식 12): + - 📌 원문 (p.293): ((6a943f06-f385-4965-81c1-6706552d7c05)) + - 📌 원문 식 (12) (p.293): [🔗 PDF 위치](((6a943ef3-97cd-4c00-9e44-307fc49d3168))) + - ![A⁻¹의 공식, 식 12 (p.293)](../assets/5장_p293_역행렬공식_식12.png) + - ⚠️ 첨자 방향 주의: $(A^{-1})_{ij} = C_{ji} / \det A$. $ij$ 성분에 $C_{ij}$ 가 아니라 $C_{ji}$ 가 들어간다. $C^{\mathrm{T}}$ 의 전치가 바로 이 뒤집힘이다 (5.1의 ⚠️ 그대로). + - 💡 5.1과의 연결: 5.1은 항등식 $AC^{\mathrm{T}} = (\det A)I$ 로 이 공식을 얻었고, 여기서는 크레이머 규칙으로 다시 얻었다. 같은 산의 두 등산로다. + - #### [8단계] 행렬식에 대한 큰 공식: n!개의 항 + - 지금까지 $\det A$ 계산법 두 가지: 여인수 공식 (5.1, 삼중대각처럼 0이 많을 때 유리), 피벗 공식 (식 7, 실전용). 세 번째가 성분 $a_{ij}$ 들만으로 직접 쓰는 **큰 공식**이다. + - 어디서 태어났나: 선형성 ❸으로 각 행을 끝까지 쪼개면 된다. $n = 2$ 로 클라이맥스까지 가보면: + - 1행 $(a, b)$ 를 $(a, 0) + (0, b)$ 로 쪼개고, 각각에서 2행 $(c, d)$ 를 $(c, 0) + (0, d)$ 로 쪼개면 행렬식 4개가 나온다: + - $$\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = \det\begin{bmatrix} a & 0 \\ c & 0 \end{bmatrix} + \det\begin{bmatrix} a & 0 \\ 0 & d \end{bmatrix} + \det\begin{bmatrix} 0 & b \\ c & 0 \end{bmatrix} + \det\begin{bmatrix} 0 & b \\ 0 & d \end{bmatrix}$$ + - 1번째와 4번째: 선택한 성분들이 같은 열에 몰려서 0인 열이 생긴다. 행렬식 0. + - 2번째: 대각행렬이라 $+ad$. 3번째: 열 교환 한 번이면 대각행렬이 되므로 $-bc$. + - 결과는 익숙한 $ad - bc$. 그리고 살아남는 조각의 조건이 보인다: **각 행에서 하나씩 고른 성분들이 서로 다른 열에 있어야 한다.** + - $n \times n$ 일반: 각 행을 $n$ 조각으로 쪼개면 $n^n$ 개 조각이 나오지만, 열이 겹치는 조각은 전부 0이다. + - 살아남는 조각: 1행에서 열 하나($n$ 가지), 2행에서 남은 열($n-1$ 가지), 계속하면 $n!$ 개. + - "각 행에서 1개, 각 열에서 1개"라는 선택 패턴은 정확히 치환행렬 $P$ 의 **1의 위치**다. 그래서 치환행렬이 $n!$ 개인 것과 항이 $n!$ 개인 것이 같은 말이 된다. + - 치환행렬 하나가 항 $D_P$ 하나를 만든다: + - 곱: $P$ 의 1 위치에 있는 $A$ 의 성분 $n$ 개를 곱한다 ($a_{1\alpha} a_{2\beta} \cdots a_{n\omega}$). + - 부호: $\det P = \pm 1$ 을 인수로 붙인다. $P$ 가 짝수 치환(행 교환 짝수 번으로 $I$ 가 됨)이면 $+1$, 홀수 치환이면 $-1$. + - 큰 공식 (식 13): 행렬식은 이 $n!$ 개 항의 합이다. + - 📌 원문 식 (13) (p.294): [🔗 PDF 위치](((6a944680-4b3c-4e0c-8151-af9190aef983))) + - ![큰 공식, 식 13 (p.294)](../assets/5장_p294_큰공식_식13.png) + - 🔢 $n = 3$: 치환이 $3! = 6$ 개라 항도 6개다. + - 📌 원문 6개 항 (p.294): [🔗 PDF 위치](((6a944693-c4eb-4ff1-95d6-796a200430e3))) + - ![3×3 큰 공식의 6개 항 (p.294)](../assets/5장_p294_큰공식_3x3_6항.png) + - ![큰 공식: 치환 6개 = 항 6개](../assets/5장_5.2_큰공식_치환6개.png) + - 파랑 3개(짝수 치환)가 $+aqz, +brx, +cpy$. 빨강 3개(홀수 치환)가 $-ary, -bpz, -cqx$. + - 💡 5.1 식 (1)과 동일한 공식이다: 5.1에서 "행 방향 화살표는 +, 역방향은 −"로 기억한 6개 항이 바로 이것. + - 부호의 진짜 정체는 화살표 방향이 아니라 치환의 짝홀이다. $4 \times 4$ 부터는 화살표 그림이 통하지 않고 치환의 짝홀만 통한다 (연습문제 16). + - 💡 여인수 공식과의 관계: 큰 공식에서 1행 성분별로 항을 묶으면 여인수 공식이 나온다. + - $+aqz - ary = a(qz - ry) = aC_{11}$ + - $+brx - bpz = b(rx - pz) = bC_{12}$ + - $+cpy - cqx = c(py - qx) = cC_{13}$ + - 즉 여인수 $C_{1j}$ 는 "1행에서 $a_{1j}$ 를 고른 모든 치환의 나머지 부분을 모은 것"이다. 여인수 공식, 피벗 공식, 큰 공식은 같은 값을 다르게 조직한 세 가지 방법일 뿐이다. + - ⚠️ 실용성: $n!$ 은 폭발한다 ($10! = 3{,}628{,}800$). 큰 공식은 계산용이 아니라 "행렬식이 무엇인지"를 보여주는 이론용이다. 계산은 피벗(식 7)으로. +- ### ❓ 스스로 점검 + - $3 \times 3$ 행렬 $A$ 의 행렬식이 5일 때, $\det(2A)$ 는? #card + id:: 6a9445f5-4d01-40e3-907f-03b3a48b60bc + - 40이다. + - $2A$ 는 행 3개 각각에 2를 곱한 행렬이므로, 행마다 선형성(❸)이 한 번씩 적용되어 $2^3 \times 5$. + - 일반화: $\det(cA) = c^n \det A$. + - $\det(A + B) = \det A + \det B$ 는 성립하는가? #card + id:: 6a9445f5-54ce-490d-ba1e-fcb6a0dcb789 + - 성립하지 않는다. + - 행렬식은 행렬 전체에 선형이 아니라 "행 하나씩"에만 선형이다. + - 반례: $I + I = 2I$ 의 행렬식은 $2^n$ 이지 $1 + 1 = 2$ 가 아니다 ($n \geq 2$). + - 직교행렬 $Q$ 의 행렬식이 1 또는 −1인 이유는? #card + id:: 6a9445f5-0671-4880-bce0-ed5ea6bce528 + - $Q^{\mathrm{T}}Q = I$ 의 양변에 행렬식을 취하면 $(\det Q^{\mathrm{T}})(\det Q) = 1$. + - $\det Q^{\mathrm{T}} = \det Q$ 이므로 $(\det Q)^2 = 1$. + - (행 교환이 없는) 소거가 행렬식을 바꾸지 않는 이유를 한 줄로 말하면? #card + id:: 6a9445f5-cab9-4eda-83fb-e324cd0d7cfd + - 행 빼기를 선형성으로 쪼개면 $\det A - d \cdot (\text{같은 행 2개인 행렬식})$ 인데, 뒤 항이 0이라 $\det A$ 만 남는다. + - $\det(ABC)$ 는? #card + id:: 6a9445f5-9876-45e1-a939-2e52ebbf245d + - $(\det A)(\det B)(\det C)$. + - 곱 규칙을 두 번 쓴다: $\det(ABC) = \det(AB) \cdot \det C = (\det A)(\det B)(\det C)$. + - 크레이머 규칙에서 $B_j$ 는 어떤 행렬이고, 규칙을 쓸 수 있는 조건은? #card + id:: 6a9445f5-5dfa-493c-8612-76b1a5d14ee5 + - $B_j$ 는 $A$ 의 $j$ 열을 $b$ 로 바꾼 행렬이다. + - 조건은 $\det A \neq 0$ (분모가 되므로). + - $\det A = 0$ 이면: $\det B_j \neq 0$ 인 것이 하나라도 있으면 해가 없고, 전부 0이면 $0/0$ 부정이다 (연습문제 7). + - 행렬 식의 큰 공식의 항이 정확히 $n!$ 개인 이유는? #card + id:: 6a9445f5-762f-4cae-978b-b0e7f2e566d6 + - 각 항은 행마다 성분 1개씩 고르는데, 열이 겹치면 그 조각은 0이라 사라진다. + - 살아남는 선택은 "각 행에서 1개, 각 열에서 1개", 즉 열의 순열이므로 $n!$ 개다. + - $A^{-1}$ 공식에서 $(A^{-1})_{ij}$ 에 $C_{ij}$ 가 아니라 $C_{ji}$ 가 들어가는 이유는? #card + id:: 6a9445f5-17fc-457a-9018-b0129601b12e + - + - $A^{-1} = C^{\mathrm{T}} / \det A$ 이기 때문이다. + - 전치가 첨자 $ij$ 를 $ji$ 로 뒤집는다. +- ### ✏️ 연습문제 풀이 (솔루션 참고) + - **문제 1** (p.294): $\det A = 2$ 일 때 $\det(A^{-1})$, $\det(A^n)$, $\det(A^{\mathrm{T}})$ 를 구하라. + - 🔁 개념 상기: 곱 규칙(식 3)과 전치 규칙(식 2)만으로 끝나는 문제다. + - 풀이: + - $\det(A^{-1})$: $AA^{-1} = I$ 에 곱 규칙을 적용하면 $(\det A)(\det A^{-1}) = \det I = 1$. 따라서 $\det A^{-1} = 1/\det A = 1/2$. + - $\det(A^n)$: 곱 규칙을 반복하면 $(\det A)^n = 2^n$. + - $\det(A^{\mathrm{T}})$: 식 (2)에 의해 $\det A = 2$ 그대로. + - 검산: $n = -1$ 을 두 번째 답에 넣으면 $2^{-1} = 1/2$ 로 첫 번째 답과 맞물린다. + - 핵심 문장: $\det(A^{-1}) = 1/\det A$, $\det(A^n) = (\det A)^n$, $\det(A^{\mathrm{T}}) = \det A$. + - **문제 2** (p.294): 행렬 $A, B, C, D$ 의 행렬식을 구하라. 이 행렬들의 열들은 선형독립인가? + - $$A = \begin{bmatrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}, \quad C = \begin{bmatrix} A & 0 \\ 0 & A \end{bmatrix}, \quad D = \begin{bmatrix} A & 0 \\ 0 & B \end{bmatrix}$$ + - 🔁 개념 상기: 판정 하나면 된다. $\det \neq 0$ 이면 열들이 선형독립, $\det = 0$ 이면 종속. 블록대각행렬은 블록별 행렬식의 곱이다. + - 풀이: + - $\det A$: 1행 여인수 전개로 $1 \cdot (0 - 1) - 1 \cdot (1 - 0) + 0 = -2$. 0이 아니므로 열들은 **선형독립**. + - $\det B$: 2행에서 1행을 빼면 $(3, 3, 3)$, 3행에서 2행을 빼면 역시 $(3, 3, 3)$. 같은 행이 2개가 되므로 $\det B = 0$. 열들은 **선형종속**. + - $\det C$: 블록대각이므로 $(\det A)(\det A) = (-2)^2 = 4 \neq 0$. **선형독립**. + - 블록대각의 행렬식이 곱이 되는 이유: 소거가 두 블록 안에서 따로 진행되어 피벗 목록이 "$A$ 의 피벗들 두 벌"이 되기 때문. + - $\det D$: 같은 이유로 $(\det A)(\det B) = (-2) \cdot 0 = 0$. **선형종속** ($B$ 부분의 종속 관계가 $D$ 안에서 그대로 산다). + - ⚠️ 해답집 주의: 해답집은 $\det C = -1$ 로 되어 있는데, 원서의 $C$ 가 한국어판과 다른 행렬로 보인다. 한국어판에 인쇄된 $C = \begin{bmatrix} A & 0 \\ 0 & A \end{bmatrix}$ 기준 정답은 4다 (5.1 문제 6과 같은 유형의 판본 불일치). + - 핵심 문장: $\det\begin{bmatrix} A & 0 \\ 0 & B \end{bmatrix} = (\det A)(\det B)$, 그리고 $\det = 0$ 판정이 곧 열의 종속 판정이다. + - **문제 3** (p.295): $x$ 로 표기한 5개의 수와 상관없이 $\det A = 0$ 임을 보여라. 1행에 대한 여인수는? 랭크는? 6개 항은? + - $$A = \begin{bmatrix} x & x & x \\ 0 & 0 & x \\ 0 & 0 & x \end{bmatrix}$$ + - 🔁 개념 상기: 같은 결론($\det = 0$)을 세 가지 도구(여인수 공식, 같은 행/랭크, 큰 공식)로 확인하는 문제다. + - 풀이 ① 여인수 공식: 1행의 여인수 3개가 전부 0이다. + - $C_{11} = +\begin{vmatrix} 0 & x \\ 0 & x \end{vmatrix} = 0$ (1열이 0), $C_{12} = -\begin{vmatrix} 0 & x \\ 0 & x \end{vmatrix} = 0$, $C_{13} = +\begin{vmatrix} 0 & 0 \\ 0 & 0 \end{vmatrix} = 0$. + - 따라서 $\det A = x C_{11} + x C_{12} + x C_{13} = 0$. 1행에 무엇이 있어도 0이다. + - 풀이 ② 랭크: 2행과 3행이 $(0, 0, x)$ 로 동일하다. 같은 행이 2개면 행렬식 0. 랭크로 말하면 서로 다른 행이 최대 2개라 랭크 $\leq 2 < 3$. + - 풀이 ③ 큰 공식: 각 항은 2행과 3행에서 서로 다른 열을 골라야 하는데, 두 행의 0 아닌 성분은 3열에만 있다. 어떤 항이든 0 인수를 포함하므로 6개 항 전부 0이다. + - 관찰 (해답집): 소거 관점으로는 "2열에 피벗이 없다"가 같은 이야기다. + - 핵심 문장: 행렬식이 0인 것은 한 가지 사실의 세 얼굴이다: 여인수 전부 0, 같은 행 2개(랭크 부족), 모든 항에 0 인수. + - **문제 5** (p.295): 크레이머 규칙 $x_j = \det B_j / \det A$ 로 다음을 풀어라. + - 🔁 개념 상기: $B_j$ 는 $A$ 의 $j$ 열을 $b$ 로 바꾼 행렬. 행렬식 $(n+1)$ 개를 만들고 나누면 끝. + - (a) $2x_1 + 5x_2 = 1$, $x_1 + 4x_2 = 2$: + - $\det A = \begin{vmatrix} 2 & 5 \\ 1 & 4 \end{vmatrix} = 8 - 5 = 3$ + - $\det B_1 = \begin{vmatrix} 1 & 5 \\ 2 & 4 \end{vmatrix} = 4 - 10 = -6$, $\det B_2 = \begin{vmatrix} 2 & 1 \\ 1 & 2 \end{vmatrix} = 4 - 1 = 3$ + - $x_1 = -6/3 = -2$, $x_2 = 3/3 = 1$. + - 검산: $2(-2) + 5(1) = 1$, $(-2) + 4(1) = 2$. 일치. + - (b) $2x_1 + x_2 = 1$, $x_1 + 2x_2 + x_3 = 0$, $x_2 + 2x_3 = 0$: + - $\det A = \begin{vmatrix} 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2 \end{vmatrix} = 2(4 - 1) - 1(2 - 0) + 0 = 4$ + - $\det B_1 = \begin{vmatrix} 1 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 1 & 2 \end{vmatrix} = 1 \cdot (4 - 1) = 3$ (1열 전개) + - $\det B_2 = \begin{vmatrix} 2 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 0 & 2 \end{vmatrix} = 2 \cdot \begin{vmatrix} 2 & 1 \\ 1 & 0 \end{vmatrix} = 2(0 - 1) = -2$ (3행 전개) + - $\det B_3 = \begin{vmatrix} 2 & 1 & 1 \\ 1 & 2 & 0 \\ 0 & 1 & 0 \end{vmatrix} = 1 \cdot \begin{vmatrix} 1 & 2 \\ 0 & 1 \end{vmatrix} = 1$ (3열 전개, 부호 $(-1)^{1+3} = +$) + - $x_1 = 3/4$, $x_2 = -2/4 = -1/2$, $x_3 = 1/4$. + - 검산: $2(3/4) + (-1/2) = 1$, $3/4 + 2(-1/2) + 1/4 = 0$, $(-1/2) + 2(1/4) = 0$. 모두 일치 (해답집 일치). + - 핵심 문장: 크레이머 규칙 사용법은 "$j$ 열을 $b$ 로 갈아끼운 행렬식을 $\det A$ 로 나눈다" 하나뿐이다. + - **문제 7** (p.295): $\det A = 0$ 이면 크레이머 규칙을 쓸 수 없다. (a)는 해가 없고 (b)는 해가 무수히 많다. 각 경우 $x_j = \det B_j / \det A$ 는 무엇인가? + - 🔁 개념 상기: 크레이머 규칙의 전제는 $\det A \neq 0$. 전제가 무너질 때 "어떻게 실패하는지"가 해의 상황을 그대로 보여준다. + - (a) $2x_1 + 3x_2 = 1$, $4x_1 + 6x_2 = 1$ (평행한 직선): + - $\det A = 12 - 12 = 0$, $\det B_1 = \begin{vmatrix} 1 & 3 \\ 1 & 6 \end{vmatrix} = 3$, $\det B_2 = \begin{vmatrix} 2 & 1 \\ 4 & 1 \end{vmatrix} = -2$ + - $x_1 = 3/0$, $x_2 = -2/0$: 정의되지 않는다. **해 없음**과 대응. + - (b) $2x_1 + 3x_2 = 1$, $4x_1 + 6x_2 = 2$ (동일한 직선): + - $\det B_1 = \begin{vmatrix} 1 & 3 \\ 2 & 6 \end{vmatrix} = 0$, $\det B_2 = \begin{vmatrix} 2 & 1 \\ 4 & 2 \end{vmatrix} = 0$ + - $x_1 = x_2 = 0/0$: 부정. **해가 무수히 많음**과 대응. + - 💡 "0 아닌 수를 0으로 나눔 = 모순(해 없음)", "$0/0$ = 정보 없음(해 무한)". 실패하는 모양조차 해의 개수를 알려준다. + - 핵심 문장: $\det A = 0$ 일 때 $\det B_j \neq 0$ 이 있으면 해 없음, 전부 0이면 부정(해 무한). + - **문제 10** (p.296): 대각선 3개가 1, 1, 1인 $n \times n$ 삼중대각행렬 $E_n$ 에 대해 $E_n = E_{n-1} - E_{n-2}$ 임을 보이고, $E_1 = 1$, $E_2 = 0$ 에서 $E_3, \ldots, E_8$ 을 구한 뒤 규칙을 관찰해 $E_{100}$ 을 구하라. + - 🔁 개념 상기: 5.1 문제 13(−1, 2, −1 행렬)과 똑같은 "1행 여인수 전개로 점화식 만들기" 패턴이다. + - [1] 점화식 유도 (1행 전개): + - $a_{11} = 1$ 의 여인수: 1행 1열을 지우면 크기 $n-1$ 의 같은 삼중대각행렬. 기여 $+1 \cdot E_{n-1}$. + - $a_{12} = 1$ 의 여인수: 부호 $(-1)^{1+2} = -1$. 1행 2열을 지운 행렬의 1열은 $(1, 0, \ldots, 0)$ 이므로 1열 전개하면 $E_{n-2}$ 만 남는다. 기여 $-1 \cdot 1 \cdot E_{n-2}$. + - 따라서 $E_n = E_{n-1} - E_{n-2}$. + - [2] 수열 계산: + - $E_1 = 1$, $E_2 = 0$ + - $E_3 = 0 - 1 = -1$, $E_4 = -1 - 0 = -1$, $E_5 = -1 - (-1) = 0$, $E_6 = 0 - (-1) = 1$ + - $E_7 = 1 - 0 = 1 = E_1$, $E_8 = 1 - 1 = 0 = E_2$ + - [3] 규칙: $1, 0, -1, -1, 0, 1$ 이 **주기 6**으로 반복된다. + - $E_7 = E_1$, $E_8 = E_2$ 이고 점화식이 직전 두 항만 보므로, 이후 전체가 그대로 반복된다: $E_{n+6} = E_n$. + - [4] $E_{100}$: $100 = 6 \times 16 + 4$ 이므로 $E_{100} = E_4 = -1$ (해답집 일치). + - 검산: $E_3 = \begin{vmatrix} 1 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & 1 & 1 \end{vmatrix} = 1(1-1) - 1(1-0) + 0 = -1$. 점화식 값과 일치. + - 💡 5.1의 −1, 2, −1 행렬은 $\det A_n = n + 1$ 로 커져 갔는데, 1, 1, 1 행렬은 주기 6으로 맴돈다. 점화식의 부호 하나가 운명을 가른다. + - 핵심 문장: 1, 1, 1 삼중대각행렬식은 $E_n = E_{n-1} - E_{n-2}$ 를 따르며 $1, 0, -1, -1, 0, 1$ 주기 6으로 반복된다. + - **문제 13** (p.296): 행렬 $A$ 를 $U$ 로 소거한 뒤 피벗의 곱으로 $\det A$ 를 구하라. + - 🔁 개념 상기: 식 (7). 행 교환 없이 소거했다면 행렬식은 그대로이므로 피벗만 곱하면 된다. + - (a) $A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{bmatrix}$: + - 소거: 2행 $-$ 1행, 3행 $-$ 1행 $\to$ 2행 $(0, 1, 1)$, 3행 $(0, 1, 2)$. 다시 3행 $-$ 2행 $\to$ $(0, 0, 1)$. + - $U = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}$, 피벗 $1, 1, 1$. 따라서 $\det A = 1$. + - 검산 (여인수): $1(6-4) - 1(3-2) + 1(2-2) = 2 - 1 + 0 = 1$. 일치. + - (b) $A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 2 & 3 \\ 3 & 3 & 3 \end{bmatrix}$: + - 소거: 2행 $- 2 \times$ 1행 $\to (0, -2, -3)$, 3행 $- 3 \times$ 1행 $\to (0, -3, -6)$. 다시 3행 $- \frac{3}{2} \times$ 2행 $\to (0, 0, -\frac{3}{2})$. + - 피벗 $1, -2, -\frac{3}{2}$. 따라서 $\det A = 1 \times (-2) \times (-\frac{3}{2}) = 3$. + - 검산 (여인수): $1(6-9) - 2(6-9) + 3(6-6) = -3 + 6 + 0 = 3$. 일치. + - 💡 분수 피벗이 나와도 곱하면 깔끔한 값이 된다. 행 교환을 했다면 교환 횟수만큼 $-1$ 을 곱해야 한다는 것만 기억. + - 핵심 문장: 행 교환이 없으면 $\det A = $ 피벗들의 곱, 행 교환이 있으면 $\pm$ 이 붙는다. + - **문제 16** (p.296): $4 \times 4$ 큰 공식의 항 $\pm a_{12}a_{21}a_{34}a_{43}$, $\pm a_{13}a_{22}a_{31}a_{44}$, $\pm a_{11}a_{22}a_{34}a_{43}$ 에 대해 $\det P$ 가 $+1$ 인지 $-1$ 인지 결정하라. + - 🔁 개념 상기: 항 $a_{1j_1}a_{2j_2}a_{3j_3}a_{4j_4}$ 의 부호는 열 첨자 순서 $(j_1, j_2, j_3, j_4)$ 를 $(1, 2, 3, 4)$ 로 만드는 데 필요한 교환 횟수의 짝홀이다. + - (a) $a_{12}a_{21}a_{34}a_{43}$: 열 순서 $(2, 1, 4, 3)$. + - $2 \leftrightarrow 1$, $4 \leftrightarrow 3$ 교환 2번 (짝수). $\det P = +1$. + - (b) $a_{13}a_{22}a_{31}a_{44}$: 열 순서 $(3, 2, 1, 4)$. + - $3 \leftrightarrow 1$ 교환 1번 (홀수). $\det P = -1$. + - (c) $a_{11}a_{22}a_{34}a_{43}$: 열 순서 $(1, 2, 4, 3)$. + - $4 \leftrightarrow 3$ 교환 1번 (홀수). $\det P = -1$. + - 검산: 교환 순서를 다르게 잡아도 횟수의 짝홀은 변하지 않는다 (예: (a)를 4번 교환으로 정렬해도 짝수). 해답집 일치. + - 핵심 문장: 큰 공식 항의 부호 = 열 첨자 순열을 정렬하는 교환 횟수가 짝수면 $+$, 홀수면 $-$. \ No newline at end of file diff --git a/Major Study/pages/5장 행렬식.md b/Major Study/pages/5장 행렬식.md index 3210b9a..557ad75 100755 --- a/Major Study/pages/5장 행렬식.md +++ b/Major Study/pages/5장 행렬식.md @@ -16,3 +16,4 @@ deck:: Logseq/전공공부/선형대수학 - 💡 수위 조절: 서론이 던지는 크레이머 규칙, 에지행렬, 부피 이야기는 전부 5.2~5.3의 예고편이다. 지금은 "행렬식 = 행렬을 요약하는 놀라운 수 하나, 계산법은 3가지"라는 감만 잡으면 충분하다. - ## 소단원 - [[5.1 3×3 행렬식과 여인수]] + - [[5.2 행렬식의 계산과 응용]] diff --git a/Major Study/pages/hls__5장_행렬식_1788006854821_0.md b/Major Study/pages/hls__5장_행렬식_1788006854821_0.md index b3a306d..3fa6301 100755 --- a/Major Study/pages/hls__5장_행렬식_1788006854821_0.md +++ b/Major Study/pages/hls__5장_행렬식_1788006854821_0.md @@ -125,4 +125,191 @@ file-path:: ../assets/5장_행렬식_1788006854821_0.pdf ls-type:: annotation hl-page:: 8 hl-color:: yellow - id:: 6a93173e-bc93-4d9b-b218-aac33dee3dde \ No newline at end of file + id:: 6a93173e-bc93-4d9b-b218-aac33dee3dde +- 가역행렬 A는 detA ≠ 0 + ls-type:: annotation + hl-page:: 11 + hl-color:: yellow + id:: 6a9439f3-8c77-4ce4-bf40-de233cd12495 +- 특이행렬 A는 detA = 0 + ls-type:: annotation + hl-page:: 11 + hl-color:: yellow + id:: 6a943a00-fc7a-40ae-8856-7a6371a40c25 +- Ax = λx에서 고윳값 λ와 고유벡터 x는 고윳값 방정식 (A−λI)x = 0을 만족한다. 즉 A−λI가 특이행렬이고 det(A−λI) = 0 + ls-type:: annotation + hl-page:: 11 + hl-color:: yellow + id:: 6a943a60-ddf1-47f7-93fb-85ab1f858a28 +- 행렬이 삼각행렬이나 대각행렬이라면, 다음과 같이 단순히 주대각성분들을 곱함으로써 행렬식을 계산 + ls-type:: annotation + hl-page:: 11 + hl-color:: yellow + id:: 6a943a8d-513d-4a41-842f-dc51c4309ade +- [:span] + ls-type:: annotation + hl-page:: 11 + hl-color:: yellow + id:: 6a943a93-ffe5-493c-beec-c2e373dfec63 + hl-type:: area + hl-stamp:: 1788099218739 +- [:span] + ls-type:: annotation + hl-page:: 11 + hl-color:: yellow + id:: 6a943a9f-5ef4-4a1a-aeae-6b7556fdeb58 + hl-type:: area + hl-stamp:: 1788099231645 +- [:span] + ls-type:: annotation + hl-page:: 11 + hl-color:: yellow + id:: 6a943aa6-c986-4f50-b6ae-fa2af53fff53 + hl-type:: area + hl-stamp:: 1788099238051 +- 직교행렬 Q의 행렬식은 1 또는 -1이다. + ls-type:: annotation + hl-page:: 11 + hl-color:: yellow + id:: 6a943ace-dea6-4c62-bf04-eb09243c01ef +- [:span] + ls-type:: annotation + hl-page:: 12 + hl-color:: yellow + id:: 6a943b00-83df-4cd4-b047-1f23ac2d8d6a + hl-type:: area + hl-stamp:: 1788099328536 +- A = LU이면 detA = (detL)(detU) = detU이다. 삼각행렬 U의 행렬식은 피벗들의 곱 + ls-type:: annotation + hl-page:: 12 + hl-color:: yellow + id:: 6a943b6f-c65a-4e0b-b6e8-afae671d15f7 +- PA = LU이면 행 교환이 존재하므로 detP = 1 또는 -1이다(치환행렬 + ls-type:: annotation + hl-page:: 12 + hl-color:: yellow + id:: 6a943b74-de8b-439f-80f0-6a3e074bc78b +- [:span] + ls-type:: annotation + hl-page:: 12 + hl-color:: yellow + id:: 6a943b90-0266-4228-baf4-227d765daec4 + hl-type:: area + hl-stamp:: 1788099472392 +- 행렬 A의 두 행이 동일하면 A의 행렬식은 0이다 + ls-type:: annotation + hl-page:: 12 + hl-color:: yellow + id:: 6a943be0-7639-45fc-ae5e-fa811c704e4f +- j행에서 i행에 d를 곱한 것을 빼도 detA는 변하지 않는다 + ls-type:: annotation + hl-page:: 13 + hl-color:: yellow + id:: 6a943c8c-83fb-4053-afa3-f1a2b63d5179 +- [:span] + ls-type:: annotation + hl-page:: 13 + hl-color:: yellow + id:: 6a943c91-84b2-4cc8-86de-b1018447eac9 + hl-type:: area + hl-stamp:: 1788099729531 +- (행 교환이 없는) 소거 과정은 행렬식을 바꾸지 않고 A를 간단한 형태로 만든다 + ls-type:: annotation + hl-page:: 13 + hl-color:: yellow + id:: 6a943cf9-6d55-4024-ba3b-d3a3996e7755 +- [:span] + ls-type:: annotation + hl-page:: 13 + hl-color:: yellow + id:: 6a943d04-373b-4865-b6d7-26b3b6f77324 + hl-type:: area + hl-stamp:: 1788099843701 +- [:span] + ls-type:: annotation + hl-page:: 13 + hl-color:: yellow + id:: 6a943d32-0ad3-46e3-bc58-5936bbb3736c + hl-type:: area + hl-stamp:: 1788099890542 +- 항등행렬 I의 1열을 x로 바꾼다. 이렇게 바꾼 삼각행렬 M₁의 행렬식은 x₁이다. M₁에 행렬 A를 곱하면 1열은 Ax가 되고, 가정에 따라 Ax = b이다. B₁의 나머지 열은 행렬 A와 동일하다. + ls-type:: annotation + hl-page:: 13 + hl-color:: yellow + id:: 6a943dc0-8127-417c-9142-7f364dec2cbc +- [:span] + ls-type:: annotation + hl-page:: 13 + hl-color:: yellow + id:: 6a943dc5-2624-4f08-b3e4-b2f6a1d99dc2 + hl-type:: area + hl-stamp:: 1788100037189 +- [:span] + ls-type:: annotation + hl-page:: 14 + hl-color:: yellow + id:: 6a943dfb-3873-4e19-a53f-6bd6e3d16dc2 + hl-type:: area + hl-stamp:: 1788100091108 +- [:span] + ls-type:: annotation + hl-page:: 14 + hl-color:: yellow + id:: 6a943e05-9a21-41e1-b292-8eb0ff3c1eb4 + hl-type:: area + hl-stamp:: 1788100100768 +- [:span] + ls-type:: annotation + hl-page:: 14 + hl-color:: yellow + id:: 6a943e84-a662-40ad-9401-d69a0546bb42 + hl-type:: area + hl-stamp:: 1788100227948 +- [:span] + ls-type:: annotation + hl-page:: 15 + hl-color:: yellow + id:: 6a943ef3-97cd-4c00-9e44-307fc49d3168 + hl-type:: area + hl-stamp:: 1788100339170 +- 크레이머 규칙에서 각 Bⱼ의 행렬식은 행렬 A의 여인수이다. + ls-type:: annotation + hl-page:: 15 + hl-color:: yellow + id:: 6a943efb-271f-4733-ab93-21fa9bd6557c +- A⁻¹는 행렬 A의 여인수들을 포함한다 + ls-type:: annotation + hl-page:: 15 + hl-color:: yellow + id:: 6a943f06-f385-4965-81c1-6706552d7c05 +- D(A) = detAB/detB가 앞서 살펴본 성질 ❶, ❷, ❸을 만족함을 보이고자 한다 + ls-type:: annotation + hl-page:: 13 + hl-color:: yellow + id:: 6a94462d-4af3-413d-b22b-6911e58b9b2c +- [:span] + ls-type:: annotation + hl-page:: 14 + hl-color:: yellow + id:: 6a94464a-ed55-45c7-96d4-135fa44a9387 + hl-type:: area + hl-stamp:: 1788102217715 +- 마지막 4개의 행렬식의 값은 각각 d, -c, -b, a이다(이들은 모두 여인수이다) + ls-type:: annotation + hl-page:: 15 + hl-color:: yellow + id:: 6a944666-5084-47c1-b9a8-0960750738ea +- [:span] + ls-type:: annotation + hl-page:: 16 + hl-color:: yellow + id:: 6a944680-4b3c-4e0c-8151-af9190aef983 + hl-type:: area + hl-stamp:: 1788102271902 +- [:span] + ls-type:: annotation + hl-page:: 16 + hl-color:: yellow + id:: 6a944693-c4eb-4ff1-95d6-796a200430e3 + hl-type:: area + hl-stamp:: 1788102291398 \ No newline at end of file

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