From 419e8f78308bdcb0e257e20d6ef53739588c4e66 Mon Sep 17 00:00:00 2001 From: songyc macbook Date: Tue, 11 Aug 2026 20:40:55 +0900 Subject: [PATCH] =?UTF-8?q?4.2=20=EC=A7=81=EC=84=A0=EA=B3=BC=20=EB=B6=80?= =?UTF-8?q?=EB=B6=84=EA=B3=B5=EA=B0=84=20=EC=9C=84=EB=A1=9C=EC=9D=98=20?= =?UTF-8?q?=EC=82=AC=EC=98=81:=20=EC=86=8C=EB=8B=A8=EC=9B=90=20=EC=A0=95?= =?UTF-8?q?=EB=A6=AC=20(=EC=82=AC=EC=9A=A9=EC=9E=90=20=EB=A9=94=EB=AA=A8?= =?UTF-8?q?=203=EA=B1=B4=20=EC=A4=91=EC=A0=90=20=EB=B0=98=EC=98=81)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit - 6단계 재구성: 사영 정의(점 관점 PB 벡터 해석) → 좌표축 사영·직교여공간 재조명 → 직선 사영 유도 → P와 I−P → 정규방정식 유도 상세 → AᵀA 가역성 - 하이라이트 24개 연결(영역 7개 이미지 추출), OCR 오타 17건 교정 - 그림: 그림 4.2·4.3·4.4·4.5 추출 + 자작 SVG 2개(직교여공간 vs 그냥 직교, P와 I−P 두 사영) - I−P 오해 교정(a를 b에 사영 아님) ⚠️ 블록 추가 - 연습문제 1·8·9·14·17·18·20·21 표준 구조 풀이 - 4장 페이지: 4.2 링크 추가, 요구사항 메모 삭제 Co-Authored-By: Claude Fable 5 Claude-Session: https://claude.ai/code/session_01Ufr6wUhJAzBaqcaBiRJ1xk 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b를 남김없이 나눠 가짐) + 둘 다 "b를" 사영한다. 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483.75647735595703, + :width 718, + :height 1015.4567293154772} + {:x1 73.51345317065716, + :y1 497.3741226196289, + :x2 429.1558024138212, + :y2 510.5685501098633, + :width 718, + :height 1015.4567293154772}), + :page 13}, + :content {:text "벡터 b 는 S 안에서 가장 가까운 점 p 로 사영된다 . 그러면 오차 벡터 e =b-p 는 부분 공간 S와 직교 한다. "}, + :properties {:color "yellow"}} + {:id #uuid "6a7a0403-2799-49a5-952d-9fdf5d50e671", + :page 13, + :position {:bounding {:x1 73.51345317065716, + :y1 497.3741226196289, + :x2 622.670878097415, + :y2 567.3393936157227, + :width 718, + :height 1015.4567293154772}, + :rects ({:x1 464.0299174040556, + :y1 497.3741226196289, + :x2 622.670878097415, + :y2 510.5685501098633, + :width 718, + :height 1015.4567293154772} + {:x1 73.51345317065716, + :y1 522.8515396118164, + :x2 617.382792159915, + :y2 539.518196105957, + :width 718, + :height 1015.4567293154772} + {:x1 73.51345317065716, + :y1 551.3671646118164, + :x2 261.3311869353056, + :y2 567.3393936157227, + :width 718, + :height 1015.4567293154772}), + :page 13}, + :content {:text "행렬 A 가 선형독립 인 열을갖는다면, 벡터 b 의 C ( A) 위로의 사영 은 P=A( ATA) 'ATb이다 . 여기서 사영 행렬 P= A( ATA) ' AT는 대칭 행렬이다"}, + :properties {:color "yellow"}} + {:id #uuid "6a7a040f-bcd4-4e55-863a-6152ed91aa7f", + :page 13, + :position {:bounding {:x1 72.51518504321575, + :y1 551.3671646118164, + :x2 619.94358317554, + :y2 595.1605606079102, + :width 718, + :height 1015.4567293154772}, + :rects ({:x1 430.3602396696806, + :y1 551.3671646118164, + :x2 619.94358317554, + :y2 567.3393936157227, + :width 718, + :height 1015.4567293154772} + {:x1 72.51518504321575, + :y1 579.1883316040039, + :x2 332.7189127653837, + :y2 595.1605606079102, + :width 718, + :height 1015.4567293154772}), + :page 13}, + :content {:text "P2P이다. 즉 두 번째 사영 은 pp로 사영하므로 아무것도 변하지 않는다"}, + :properties {:color "yellow"}} + {:id #uuid "6a7a0415-779e-42a1-9b91-95340aa2264b", + :page 13, + :position {:bounding {:x1 87.36979167163372, + :y1 741.3844985961914, + :x2 616.7273356169462, + :y2 782.0637588500977, + :width 718, + :height 1015.4567293154772}, + :rects ({:x1 87.36979167163372, + :y1 741.3844985961914, + :x2 616.7273356169462, + :y2 755.2734146118164, + :width 718, + :height 1015.4567293154772} + {:x1 87.36979167163372, + :y1 768.1749038696289, + :x2 576.0654703825712, + :y2 782.0637588500977, + :width 718, + :height 1015.4567293154772}), + :page 13}, + :content {:text "벡터 b 가 직선 위로 사영되면, 사영 p는 벡터 b 의 해당 직선 방향 부분 이 된다 . 벡터 b가 평면위로 사영되면, 사영 p는 벡터 b의 해당 평면 방향 부분 이 된다. 사영 p는 Pb 이다."}, + :properties {:color "yellow"}} + {:id #uuid "6a7a041f-7af3-4273-a7e3-c56d9b591165", + :page 14, + :position {:bounding {:x1 86, + :y1 382.9990234375, + :x2 660, + :y2 453.9990234375, + :width 718, + :height 1015.4567293154772}, + :rects (), + :page 14}, + :content {:text "[:span]", :image 1786381341457}, + :properties {:color "yellow"}} + {:id #uuid "6a7a042f-2b0c-4ecd-949e-f28c59939ff2", + :page 14, + :position {:bounding {:x1 102.31119029223919, + :y1 710.7312927246094, + :x2 467.52057902514935, + :y2 724.6202087402344, + :width 718, + :height 1015.4567293154772}, + :rects ({:x1 102.31119029223919, + :y1 710.7312927246094, + :x2 467.52057902514935, + :y2 724.6202087402344, + :width 718, + :height 1015.4567293154772}), + :page 14}, + :content {:text "직선 과 평면 은 단순히 직교 하는 것을 넘어, 서로 직교 여공간 이다"}, + :properties {:color "yellow"}} + {:id #uuid "6a7a0437-3040-4bc0-a638-f5b5d33a8f99", + :page 14, + :position {:bounding {:x1 135.98089091479778, + :y1 799.5334167480469, + :x2 623.7521158903837, + :y2 816.8945007324219, + :width 718, + :height 1015.4567293154772}, + :rects ({:x1 135.98089091479778, + :y1 799.5334167480469, + :x2 623.7521158903837, + :y2 816.8945007324219, + :width 718, + :height 1015.4567293154772}), + :page 14}, + :content {:text "벡터들은 P1 + P2= b 를 만족 하고, 행렬들은 P+P2= 1, PP2 = 0 을 만족한다 . "}, + :properties {:color "yellow"}} + {:id #uuid "6a7a043f-522f-490b-b277-c1d2a36f849c", + :page 15, + :position {:bounding {:x1 379.30771891772747, + :y1 407.34588623046875, + :x2 620.6924845427275, + :y2 421.23480224609375, + :width 718, + :height 1015.4567293154772}, + :rects ({:x1 379.30771891772747, + :y1 407.34588623046875, + :x2 620.6924845427275, + :y2 421.23480224609375, + :width 718, + :height 1015.4567293154772}), + :page 15}, + :content {:text " 6 에서 p까지 의 직선 은 벡터 4 와 직교 한다"}, + :properties {:color "yellow"}} + {:id #uuid "6a7a044a-2260-4606-b890-6f4f9333742c", + :page 16, + :position {:bounding {:x1 81, + :y1 50.4716796875, + :x2 667, + :y2 166.4716796875, + :width 718, + :height 1015.4567293154772}, + :rects (), + :page 16}, + :content {:text "[:span]", :image 1786381386523}, + :properties {:color "yellow"}} + {:id #uuid "6a7a045f-fe09-47a3-b9b0-74e54a3248ee", + :page 16, + :position {:bounding {:x1 104, + :y1 276.4716796875, + :x2 508, + :y2 317.4716796875, + :width 718, + :height 1015.4567293154772}, + :rects (), + :page 16}, + :content {:text "[:span]", :image 1786381407053}, + :properties {:color "yellow"}} + {:id #uuid "6a7a046a-3e0f-494b-b603-5a93cc3eeed4", + :page 17, + :position {:bounding {:x1 89, + :y1 126.2490234375, + :x2 604, + :y2 207.2490234375, + :width 718, + :height 1015.4567293154772}, + :rects (), + :page 17}, + :content {:text "[:span]", :image 1786381417724}, + :properties {:color "yellow"}} + {:id #uuid "6a7af20e-41d5-4040-a603-a53e5809a41f", + :page 17, + :position {:bounding {:x1 63.465708419680595, + :y1 798.9257354736328, + :x2 369.44150798022747, + :y2 812.8146209716797, + :width 718, + :height 1015.4567293154772}, + :rects ({:x1 63.465708419680595, + :y1 798.9257354736328, + :x2 369.44150798022747, + :y2 812.8146209716797, + :width 718, + :height 1015.4567293154772}), + :page 17}, + :content {:text "항등행렬 / 에 대해 행렬 - P 도 사영행렬 이어야 한다"}, + :properties {:color "yellow"}} + {:id #uuid "6a7af214-6b8f-4d35-99ab-549c781aa104", + :page 17, + :position {:bounding {:x1 65.47308604419231, + :y1 826.5407409667969, + :x2 608.8986002653837, + :y2 866.1891784667969, + :width 718, + :height 1015.4567293154772}, + :rects ({:x1 85.28645579516888, + :y1 826.5407409667969, + :x2 608.8986002653837, + :y2 839.7351989746094, + :width 718, + :height 1015.4567293154772} + {:x1 65.47308604419231, + :y1 851.6058654785156, + :x2 102.84287516772747, + :y2 866.1891784667969, + :width 718, + :height 1015.4567293154772}), + :page 17}, + :content {:text "P 가 하나의 부분공간 위로 사영 할 때 , I -P 는 그 부분 공간과 직교 하는 부분공간 위로 사영 한다."}, + :properties {:color "yellow"}} + {:id #uuid "6a7af225-e248-4100-b418-ce9559de344b", + :page 18, + :position {:bounding {:x1 106.54296366870403, + :y1 313.35718154907227, + :x2 185.54686991870403, + :y2 327.94052505493164, + :width 718, + :height 1015.4567293154772}, + :rects ({:x1 106.54296366870403, + :y1 313.35718154907227, + :x2 185.54686991870403, + :y2 327.94052505493164, + :width 718, + :height 1015.4567293154772}), + :page 18}, + :content {:text "p = Ai( 사)"}, + :properties {:color "yellow"}} + {:id #uuid "6a7af22c-331d-4718-b45f-3c7997457022", + :page 18, + :position {:bounding {:x1 106.54296366870403, + :y1 313.35718154907227, + :x2 656.6622721403837, + :y2 360.95920181274414, + :width 718, + :height 1015.4567293154772}, + :rects ({:x1 288.71528117358685, + :y1 313.35718154907227, + :x2 656.6622721403837, + :y2 327.94052505493164, + :width 718, + :height 1015.4567293154772} + {:x1 106.54296366870403, + :y1 343.598087310791, + :x2 564.8734537810087, + :y2 360.95920181274414, + :width 718, + :height 1015.4567293154772}), + :page 18}, + :content {:text "위 의 ' ( hat) 표기는 열 공간 안에서 가장 가까운 벡터 를 제공하는 최적 의 선택 임을 의미 한다 . \" = 1인 경우 , 그 최적 의 선택은 호 - 이다"}, + :properties {:color "yellow"}} + {:id #uuid "6a7af232-2329-4ae5-8b0b-f32445874c99", + :page 18, + :position {:bounding {:x1 106.54296366870403, + :y1 449.164493560791, + :x2 656.1355997771025, + :y2 487.836368560791, + :width 718, 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"6a7af245-203a-4d34-99d3-63a1f7a3bc3e", + :page 19, + :position {:bounding {:x1 60, + :y1 120.220703125, + :x2 645, + :y2 400.220703125, + :width 718, + :height 1015.4567293154772}, + :rects (), + :page 19}, + :content {:text "[:span]", :image 1786442309180}, + :properties {:color "yellow"}} + {:id #uuid "6a7af250-44f8-4e4d-8c66-e774cd33390a", + :page 19, + :position {:bounding {:x1 71.28905741870403, + :y1 639.9848022460938, + :x2 520.8550211638212, + :y2 657.3458862304688, + :width 718, + :height 1015.4567293154772}, + :rects ({:x1 71.28905741870403, + :y1 639.9848022460938, + :x2 520.8550211638212, + :y2 657.3458862304688, + :width 718, + :height 1015.4567293154772}), + :page 19}, + :content {:text "핵심 은 정규 방정식 (normal equation ) AT ( b- A² )= 0을 해결 하는 단계이다 ."}, + :properties {:color "yellow"}} + {:id #uuid "6a7af25d-f986-4721-a903-e7463d07eb9a", + :page 19, + :position {:bounding {:x1 56, + :y1 733.220703125, + :x2 645, + :y2 888.220703125, + :width 718, + :height 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Study/pages/4.2 직선과 부분공간 위로의 사영.md b/Major Study/pages/4.2 직선과 부분공간 위로의 사영.md new file mode 100755 index 0000000..226b276 --- /dev/null +++ b/Major Study/pages/4.2 직선과 부분공간 위로의 사영.md @@ -0,0 +1,217 @@ +deck:: Logseq/전공공부/선형대수학 + +- ### 핵심 개념 + - **흐름 지도**: [1단계] 사영이란 무엇인가 → [2단계] 좌표축 사영으로 감 잡기 (직교여공간 재조명) → [3단계] 직선 위로의 사영 공식 → [4단계] 사영행렬 $P$ 와 $I - P$ → [5단계] 부분공간 위로의 사영 (정규방정식) → [6단계] $A^{\mathrm{T}}A$ 의 가역성. + - #### [1단계] 사영이란 무엇인가 + - **왜 필요한가**: $b$ 가 부분공간(예: 열공간) **밖**에 있으면 $Ax = b$ 는 해가 없다. 그때 할 수 있는 최선은 부분공간 **안**에서 $b$ 와 가장 가까운 점을 찾는 것. 그 점이 **사영**(projection) $p$ 다. + - **정의**: 벡터 $b$ 의 부분공간 $\mathbf{S}$ 위로의 사영 $p$ 는 $\mathbf{S}$ 안에서 $b$ 와 가장 가까운 벡터. 이때 **오차벡터** $e = b - p$ 는 $\mathbf{S}$ 와 직교한다. + - 📌 원문 (p.219): ((6a7a03f4-57ff-42c6-9820-cc0446588b7a)) + - "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)) + - ![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. + - $e = b - p$ 는 기하와 벡터에서 배운 $\overrightarrow{PB} = b - p$ 그대로, **점** $P$ **에서 점** $B$ **로 가는 벡터**다. + - $P$ 는 부분공간 위에 있는 "수선의 발"이고, $e$ 는 발에서 $B$ 를 향해 부분공간과 수직하게 올라가는 벡터다. + - $\|e\|$ 가 바로 "점 $B$ 에서 부분공간까지의 거리"다 (책 p.228 요약에도 이 문장이 있다: $b$ 에서 $\mathbf{C}(A)$ 까지의 거리는 $\|e\|$). + - 이 그림은 차원과 무관하다: 정의에 쓰인 것이 내적(직교)뿐이라, $\mathbb{R}^4$ 이상에서도 2·3차원 직관이 그대로 성립한다. 직선/평면 대신 "부분공간"으로 단어만 바뀐다. + - #### [2단계] 좌표축 사영으로 감 잡기 + - 🔢 가장 쉬운 사영부터: $b = (2, 3, 4)$ 를 $z$ 축과 $xy$-평면 위로 사영해보자. + - $z$ 축 위로: $p_1 = (0, 0, 4)$ ($b$ 의 $z$ 성분만 남긴다). + - $xy$-평면 위로: $p_2 = (2, 3, 0)$ ($x, y$ 성분만 남긴다). + - 이 "성분만 남기기"를 해주는 행렬이 사영행렬 $P_1, P_2$: + - 📌 원문 사영행렬 박스 (p.220): [🔗 PDF 위치](((6a7a041f-7af3-4273-a7e3-c56d9b591165))) + - ![z축·xy평면 사영행렬 (p.220)](../assets/4장_p220_z축_xy평면_사영행렬_박스.png) + - ![그림 4.2 P1과 P2의 사영 (p.220)](../assets/4장_p220_그림4.2_P1_P2.png) + - (p.220, 그림 4.2) + - 관찰: $p_1 + p_2 = (2,3,4) = b$. 두 사영을 합치면 원본이 복원된다! + - 📌 원문 (p.220): ((6a7a0437-3040-4bc0-a638-f5b5d33a8f99)) + - 행렬 언어로는 $P_1 + P_2 = I$ (복원), $P_1 P_2 = 0$ (한 번 $xy$-평면에 눕힌 벡터는 $z$ 성분이 없어 $z$ 축 사영이 0). + - 💡 **직교여공간 다시 제대로** (4.1 복습 + 사영의 눈으로): 책이 "직선과 평면은 단순히 직교하는 것을 넘어 서로 직교여공간이다"라고 강조하는 이유가 여기 있다. + - 📌 원문 (p.220): ((6a7a042f-2b0c-4ecd-949e-f28c59939ff2)) + - **정의 복습**: $\mathbf{V}^{\perp}$ = $\mathbf{V}$ 와 직교하는 벡터 **전부**. 두 공간이 서로의 직교여공간이려면 ① 직교하고 ② 차원 합이 전체 공간을 채워야 한다 ($1 + 2 = 3$ ✓). + - **"그냥 직교"와 뭐가 다른가**: $z$ 축과 $x$ 축도 직교하지만 $1 + 1 = 2 < 3$ 이라 여공간이 아니다. 이 차이가 사영에서 실전 차이를 만든다: + - ![직교여공간 vs 그냥 직교](../assets/4장_4.2_직교여공간_vs_그냥직교.png) + - 여공간 쌍이면: 모든 $b$ 가 두 성분으로 남김없이 분해되고 ($p_1 + p_2 = b$), 행렬로는 $P_1 + P_2 = I$. "분해했다가 재조립"이 가능하다. + - 여공간이 아니면: $z$ 축·$x$ 축 사영을 합해도 $y$ 성분이 유실되어 $b$ 를 복원할 수 없다 ($P_1 + P_2 \neq I$). + - 4.1과의 연결: 행공간·영공간이 직교여공간 쌍이라서 $x = x_r + x_n$ 분해가 가능했던 것, 정확히 같은 구조다. **"여공간 = 분해가 되는 직교"** 로 기억하면 된다. + - ⚠️ 이 절의 목표 선언 (p.220): 어떤 직선이든, 어떤 평면이든, $\mathbb{R}^m$ 의 어떤 $n$ 차원 부분공간이든, ① 사영 $p$ 를 찾고 ② $p = Pb$ 를 만드는 사영행렬 $P$ 를 찾는 것. + - #### [3단계] 직선 위로의 사영: 공식 유도 + - 설정: $a$ 방향 직선 위에서 $b$ 와 가장 가까운 점 $p$ 를 찾는다. $p$ 는 직선 위에 있으니 $p = \hat{x}a$ (미지수는 배율 $\hat{x}$ 하나). + - 유일한 무기는 직교: $b$ 에서 $p$ 까지의 오차 $e = b - \hat{x}a$ 가 $a$ 와 직교해야 한다. + - 📌 원문 (p.221): ((6a7a043f-522f-490b-b277-c1d2a36f849c)) + - $$a \cdot (b - \hat{x}a) = 0 \quad\Rightarrow\quad \hat{x} = \frac{a^{\mathrm{T}}b}{a^{\mathrm{T}}a}$$ + - 📌 원문 식 (2) 박스 (p.222): [🔗 PDF 위치](((6a7a044a-2260-4606-b890-6f4f9333742c))) + - ![직선 사영 공식 박스 (p.222)](../assets/4장_p222_직선사영_공식_박스.png) + - 결과: 사영은 $p = \hat{x}a$. + - 📌 원문 (p.222): [🔗 PDF 위치](((6a7a045f-fe09-47a3-b9b0-74e54a3248ee))) + - ![직선 사영 p 공식 (p.222)](../assets/4장_p222_직선사영_p공식.png) + - 특별한 경우 ①: $b = a$ 이면 $\hat{x} = 1$, 사영은 자기 자신 ($Pa = a$). + - 특별한 경우 ②: $b \perp a$ 이면 $a^{\mathrm{T}}b = 0$, 사영은 $p = 0$. + - 💡 이 두 극단이 사영의 정체를 말해준다: 이미 직선 위에 있으면 그대로, 완전히 수직이면 아무것도 안 남는다. 일반 $b$ 는 그 중간. + - 🔢 EX1 (p.222): $b = (1,1,1)$ 을 $a = (1,2,2)$ 위로 사영. + - $a^{\mathrm{T}}b = 1 + 2 + 2 = 5$, $a^{\mathrm{T}}a = 1 + 4 + 4 = 9$ → $\hat{x} = \frac{5}{9}$. + - $p = \frac{5}{9}a = (\frac{5}{9}, \frac{10}{9}, \frac{10}{9})$, $e = b - p = (\frac{4}{9}, -\frac{1}{9}, -\frac{1}{9})$. + - 검산: $e^{\mathrm{T}}a = \frac{4}{9} - \frac{2}{9} - \frac{2}{9} = 0$ ✓. + - 삼각법과의 연결 (식 (3)): $b, p, e$ 는 직각삼각형을 이루고 $\|p\| = \|b\|\cos\theta$, $\|e\| = \|b\|\sin\theta$. + - 💡 그런데 $\cos\theta$ 를 직접 구하는 것보다 $\frac{a^{\mathrm{T}}b}{a^{\mathrm{T}}a}$ 계산이 훨씬 쉽다 (EX1에서 $\cos\theta = \frac{5}{3\sqrt{3}}$ 은 제곱근 투성이지만 $\hat{x} = \frac{5}{9}$ 는 깔끔). 내적이 삼각법을 대신해주는 것이 사영 공식의 실용적 가치다. + - #### [4단계] 사영행렬 $P$ 와 $I - P$ + - $p = a\hat{x} = a\frac{a^{\mathrm{T}}b}{a^{\mathrm{T}}a}$ 에서 $b$ 에 곱해지는 부분만 떼어내면 사영행렬이 나온다: + - $$P = \frac{aa^{\mathrm{T}}}{a^{\mathrm{T}}a}$$ + - 📌 원문 사영행렬 박스 (p.223): [🔗 PDF 위치](((6a7a046a-3e0f-494b-b603-5a93cc3eeed4))) + - ![직선 위로의 사영행렬 박스 (p.223)](../assets/4장_p223_직선_사영행렬_박스.png) + - 구조 읽기: 열($a$) × 행($a^{\mathrm{T}}$) ÷ 수($a^{\mathrm{T}}a$). 크기는 $m \times m$ 이지만 **랭크 1** (3장의 $uv^{\mathrm{T}}$ 꼴). 열공간이 바로 그 직선이다. + - 🔢 EX2 (p.223): $a = (1,2,2)$ 방향 직선의 사영행렬. + - $$P = \frac{1}{9}\begin{bmatrix} 1&2&2 \\ 2&4&4 \\ 2&4&4 \end{bmatrix}$$ + - EX1의 $b = (1,1,1)$ 로 검산: $Pb = \frac{1}{9}(5, 10, 10) = p$ ✓ (같은 답). + - **사영행렬의 두 성질**: + - $P^2 = P$: 두 번 사영해도 그대로다 (이미 직선 위에 있는 $p$ 를 또 사영하면 자기 자신, [3단계] 특별한 경우 ①). + - 📌 원문 (p.219): ((6a7a040f-bcd4-4e55-863a-6152ed91aa7f)) + - $P^{\mathrm{T}} = P$ (대칭). ⚠️ 연습문제 0 (한국판 추가 문제)의 교훈: $P^2 = P$ 만으로는 "사영"이고, 오차 $e$ 가 $p$ 와 직교하는 **직교사영**이 되려면 $P^{\mathrm{T}} = P$ 까지 필요하다. 이 절의 $P$ 는 둘 다 만족한다. + - $I - P$ 도 사영행렬이다: + - 📌 원문 (p.223): ((6a7af20e-41d5-4040-a603-a53e5809a41f)) / ((6a7af214-6b8f-4d35-99ab-549c781aa104)) + - $Pb = p$ 이므로 $(I - P)b = b - p = e$. 즉 $I - P$ 는 **오차 쪽을 만들어주는 사영**이다. + - $P$ 가 부분공간 $\mathbf{S}$ 위로 사영하면, $I - P$ 는 **직교여공간** $\mathbf{S}^{\perp}$ 위로 사영한다 ([2단계]의 여공간 그림 그대로: 둘이 합쳐 $I$). + - 한 그림 정리: 같은 $b$ 를 두 곳에 사영한 것이 $p$ 와 $e$ 다. 각 사영의 오차는 상대편의 사영이 된다 ($P$ 의 오차 = $e$, $I-P$ 의 오차 = $p$). + - ![P와 I−P: 같은 b의 두 사영](../assets/4장_4.2_P와_IP_두사영.png) + - ⚠️ 흔한 오해 교정: $I - P$ 를 "$a$ 를 $b$ 에 사영하는 행렬"로 착각하기 쉽다. 아니다! 두 행렬 모두 **사영당하는 벡터는 똑같이** $b$ 이고, 바뀌는 것은 **도착지**뿐이다 ($P$ 는 직선 $L$, $I-P$ 는 여공간 $L^{\perp}$). "$a$ 를 $b$ 방향에 사영"하는 행렬은 $\frac{bb^{\mathrm{T}}}{b^{\mathrm{T}}b}$ 로, $I - P$ 와는 전혀 다른 행렬이다 (연습문제 10이 정확히 이 차이를 다룬다: $a_1$ 을 $a_2$ 에 사영하고 그 결과를 다시 $a_1$ 에 사영하는 것은 $P_1 P_2$ 라는 곱이지, $I - P$ 가 아니다). + - 💡 $b$ 의 $a$ 쪽 성분은 $P$ 가, 남은 수직 성분은 $I - P$ 가 담당. $P + (I - P) = I$ 는 "성분 분해의 완전성"이다. + - #### [5단계] 부분공간 위로의 사영: 정규방정식이 태어나는 과정 + - **문제를 다시 정확히 읽기** (여기서 헷갈리기 쉽다): $\mathbb{R}^m$ 안에 선형독립인 벡터 $a_1, \ldots, a_n$ 이 있다. 목표는 "$b$ 와 가장 가까운 **선형결합** $p = \hat{x}_1 a_1 + \cdots + \hat{x}_n a_n$ 을 구하라". + - 미지수가 무엇인지부터: 벡터가 아니라 **결합 계수** $\hat{x} = (\hat{x}_1, \ldots, \hat{x}_n)$, 즉 숫자 $n$ 개다. 직선 때 $\hat{x}$ 하나였던 것이 $n$ 개로 늘었을 뿐이다. + - $a_i$ 들을 열로 넣은 행렬 $A$ 를 쓰면 선형결합은 $Ax$ (1장), 그러니 목표는 $p = A\hat{x}$: "열공간 안에서 $b$ 와 가장 가까운 점". + - 📌 원문 (p.224): ((6a7af225-e248-4100-b418-ce9559de344b)) / ((6a7af22c-331d-4718-b45f-3c7997457022)) + - 모자(hat) 기호의 뜻: $\hat{x}$ 는 "$Ax = b$ 의 해"가 아니라 (해는 없을 수 있다!) "열공간 안에서 최선을 주는 계수"라는 표시다. + - **전략은 직선 때와 완전히 같은 3단계** (S)(C)(T): ① $\hat{x}$ 을 구하고 → ② 사영 $p = A\hat{x}$ → ③ 사영행렬 $P$. + - 📌 원문 (p.224): ((6a7af232-2329-4ae5-8b0b-f32445874c99)) + - **유도, 잘게 쪼개기** (식 (4)): 쓸 수 있는 기하 사실은 단 하나, "오차 $e = b - A\hat{x}$ 는 부분공간과 직교한다". + - 📌 원문 (p.224): ((6a7af237-5421-4b5e-9aaf-f18ab9f7b198)) + - ① "부분공간 전체와 직교" ⟺ "기저 $a_1, \ldots, a_n$ 각각과 직교" (기저와 직교하면 모든 선형결합과 직교, 4.1 문제 17 보충에서 쓴 바로 그 논리). + - 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: + - $$A^{\mathrm{T}}(b - A\hat{x}) = 0$$ + - 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)) + - 💡 "갑자기 방정식이 왜 나오지?"에 대한 답: 미지수가 계수 $n$ 개인데 직교 조건이 딱 $n$ 개의 식을 주니까, 그 둘을 맞춰 놓은 $n \times n$ 연립방정식을 푸는 것이다. 마법이 아니라 "미지수 개수 = 조건 개수" 맞추기다. 계수행렬이 $A^{\mathrm{T}}A$ 인 이유도 위 ③에서 눈으로 확인된다. + - 💡 더 빠른 유도 (책의 ❶❷, 4.1의 언어로): ❶ 부분공간은 $\mathbf{C}(A)$ 다 → ❷ 오차 $e$ 는 열공간과 직교하는 공간, 즉 **좌영공간** $\mathbf{N}(A^{\mathrm{T}})$ 에 있다 → $A^{\mathrm{T}}e = 0$. 끝. (기본정리 2가 유도를 한 줄로 만든다.) + - 📌 원문 ❶❷ 박스 (p.225): [🔗 PDF 위치](((6a7af25d-f986-4721-a903-e7463d07eb9a))) + - ![정규방정식의 빠른 유도 (p.225)](../assets/4장_p225_정규방정식_유도_박스.png) + - 덤: $b$ 는 $p$ ($\mathbf{C}(A)$ 성분)와 $e$ ($\mathbf{N}(A^{\mathrm{T}})$ 성분)로 분해된다. 4.1의 $x = x_r + x_n$ 이 $\mathbb{R}^m$ 쪽 쌍(열공간·좌영공간)에서 재현된 것이 $b = p + e$ 다! + - 🧩 **결과 3공식** (식 (5), (6), (7), 꼭 기억): + - 📌 원문 3공식 박스 (p.225): [🔗 PDF 위치](((6a7af245-203a-4d34-99d3-63a1f7a3bc3e))) + - ![사영의 3공식 (p.225)](../assets/4장_p225_사영_3공식_박스.png) + - ($n \times 1$) $\hat{x}$: $A^{\mathrm{T}}A\hat{x} = A^{\mathrm{T}}b$ 의 해, $\hat{x} = (A^{\mathrm{T}}A)^{-1}A^{\mathrm{T}}b$. + - ($m \times 1$) 사영: $p = A\hat{x} = A(A^{\mathrm{T}}A)^{-1}A^{\mathrm{T}}b$. + - ($m \times m$) 사영행렬: $P = A(A^{\mathrm{T}}A)^{-1}A^{\mathrm{T}}$. + - 📌 원문 (p.219): ((6a7a0403-2799-49a5-952d-9fdf5d50e671)) + - 검산 감각: $n = 1$ 이면 $A^{\mathrm{T}}A = a^{\mathrm{T}}a$ (1×1 숫자)라서 "역행렬"이 "나누기"가 되고, [3단계]·[4단계]의 직선 공식 (식 (8))로 정확히 환원된다. 새 공식이 아니라 직선 공식의 일반화다. + - 🔢 EX3 (p.226) 전체 계산: $A = \begin{bmatrix} 1&0 \\ 1&1 \\ 1&2 \end{bmatrix}$, $b = (6, 0, 0)$. + - ① 재료 준비: $A^{\mathrm{T}}A = \begin{bmatrix} 3&3 \\ 3&5 \end{bmatrix}$, $A^{\mathrm{T}}b = (6, 0)$. + - ② 정규방정식 풀기: $3\hat{x}_1 + 3\hat{x}_2 = 6$, $3\hat{x}_1 + 5\hat{x}_2 = 0$. 빼면 $2\hat{x}_2 = -6$ → $\hat{x} = (5, -3)$. + - ③ 사영: $p = 5(1,1,1) - 3(0,1,2) = (5, 2, -1)$. 오차: $e = b - p = (1, -2, 1)$. + - ④ 검산: $e \cdot (1,1,1) = 0$ ✓, $e \cdot (0,1,2) = -2 + 2 = 0$ ✓ (오차가 두 열 모두와 직교). + - ⑤ 사영행렬까지: $(A^{\mathrm{T}}A)^{-1} = \frac{1}{6}\begin{bmatrix} 5&-3 \\ -3&3 \end{bmatrix}$, $P = \frac{1}{6}\begin{bmatrix} 5&2&-1 \\ 2&2&2 \\ -1&2&5 \end{bmatrix}$. $Pb = (5,2,-1) = p$ ✓. + - ⚠️ $P = A(A^{\mathrm{T}}A)^{-1}A^{\mathrm{T}}$ **계산 주의**: $(A^{\mathrm{T}}A)^{-1}$ 를 $A^{-1}(A^{\mathrm{T}})^{-1}$ 로 쪼개면 안 된다! + - 📌 원문 (p.226): ((6a7af267-9dca-45c0-bc40-f0a8db37093b)) / ((6a7af26d-7da7-41db-ba30-fac9454a16e0)) + - 쪼개면 $P = AA^{-1}(A^{\mathrm{T}})^{-1}A^{\mathrm{T}} = I$ 가 나와버리는데, $A$ 는 직사각행렬이라 $A^{-1}$ 자체가 존재하지 않는다 ($(BC)^{-1} = C^{-1}B^{-1}$ 는 각각이 가역인 정사각일 때만). + - 💡 뒤집어 보면, $A$ 가 진짜 가역 정사각이면 $P = I$ 가 **맞는 답**이다: 열공간이 $\mathbb{R}^m$ 전체라 모든 $b$ 의 사영이 자기 자신 (연습문제 9·24). + - #### [6단계] $A^{\mathrm{T}}A$ 는 언제 가역인가 + - 정규방정식을 "풀 수 있다"고 말하려면 $A^{\mathrm{T}}A$ 의 가역이 보장돼야 한다. 조건은 딱 하나: + - 📌 원문 (p.227): ((6a7af283-66b3-4560-ae49-96b3acabb433)) + - 🧩 $A^{\mathrm{T}}A$ **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). + - 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. + - 💡 수위 조절: 다음 절(4.3 최소제곱)이 정확히 이 장비로 싸운다. 해가 없는 $Ax = b$ (데이터 근사)에서 정규방정식 $A^{\mathrm{T}}A\hat{x} = A^{\mathrm{T}}b$ 를 풀면 "최선의 근사해"가 나온다. 이 절에서는 "사영 계산 도구"까지만 알면 충분하다. + - **한눈 요약** (핵심 아이디어 리뷰, p.228): + - 📌 원문 (p.228): [🔗 PDF 위치](((6a7af28d-1121-4a8f-94c5-3749f4744e14))) + - ![핵심 아이디어 리뷰 (p.228)](../assets/4장_p228_핵심아이디어_리뷰.png) +- ### ❓ 스스로 점검 + - 사영 $p$ 와 오차벡터 $e$ 의 정의는? $e$ 는 무엇과 직교하며, $\|e\|$ 의 기하적 의미는? + - $p$: 부분공간 안에서 $b$ 와 가장 가까운 벡터. $e = b - p$. + - $e$ 는 그 부분공간 전체와 직교한다 ($e \in \mathbf{N}(A^{\mathrm{T}})$, 좌영공간). + - $\|e\|$ 는 $b$ 의 끝점에서 부분공간까지의 거리다 (수선의 발까지의 길이). + - 직선 위로의 사영 공식 $\hat{x} = \frac{a^{\mathrm{T}}b}{a^{\mathrm{T}}a}$ 는 어떤 한 줄에서 나오는가? + - $p = \hat{x}a$ 로 놓고, 오차 $b - \hat{x}a$ 가 $a$ 와 직교한다는 조건 $a \cdot (b - \hat{x}a) = 0$ 을 풀면 된다. + - 사영행렬은 $b$ 에 곱해지는 부분을 떼어낸 $P = \frac{aa^{\mathrm{T}}}{a^{\mathrm{T}}a}$ (랭크 1). + - 정규방정식 $A^{\mathrm{T}}A\hat{x} = A^{\mathrm{T}}b$ 에서 미지수는 무엇이고, 이 방정식은 어떤 기하 사실에서 나오는가? + - 미지수는 선형결합의 계수 $\hat{x}$ ($n$ 개의 숫자). + - 기하 사실은 "오차 $b - A\hat{x}$ 가 부분공간(= 각 기저 $a_i$)과 직교한다" 하나뿐이다. + - $n$ 개의 내적 방정식 $a_i^{\mathrm{T}}(b - A\hat{x}) = 0$ 을 행렬로 묶으면 $A^{\mathrm{T}}(b - A\hat{x}) = 0$. + - $A^{\mathrm{T}}A$ 가 가역일 필요충분조건은? 증명의 핵심 한 줄은? + - $A$ 의 열이 선형독립인 것. + - 핵심: $\mathbf{N}(A^{\mathrm{T}}A) = \mathbf{N}(A)$ ($A^{\mathrm{T}}Ax = 0$ 이면 $\|Ax\|^2 = x^{\mathrm{T}}A^{\mathrm{T}}Ax = 0$). + - $I - P$ 는 무엇 위로 사영하는 행렬인가? $P^2 = P$ 인 직관적 이유는? + - $P$ 가 사영하는 부분공간의 **직교여공간** 위로 사영한다. $(I-P)b = e$. + - 이미 부분공간 위에 있는 벡터를 다시 사영하면 자기 자신이라서 $P(Pb) = Pb$, 즉 $P^2 = P$. + - 두 사영행렬이 $P_1 + P_2 = I$ 를 만족하려면 두 부분공간은 어떤 관계여야 하는가? + - 서로의 직교여공간이어야 한다 (직교 + 차원 합 = 전체). + - 그냥 직교이기만 하면 (예: $\mathbb{R}^3$ 의 $z$ 축과 $x$ 축) 사영 합이 $b$ 를 복원하지 못한다. +- ### ✏️ 연습문제 풀이 (솔루션 참고) + - **문제 1**: 벡터 $b$ 를 $a$ 방향 직선 위로 사영하고 $e \perp a$ 를 확인하라. (a) $b = (1,2,2)$, $a = (1,1,1)$ (b) $b = (1,3,1)$, $a = (-1,-3,-1)$ + - 🔁 개념 상기: 직선 사영 3종 세트는 $\hat{x} = \frac{a^{\mathrm{T}}b}{a^{\mathrm{T}}a}$ → $p = \hat{x}a$ → $e = b - p$ ([3단계]). 검산은 항상 $e^{\mathrm{T}}a = 0$. + - (a) 계산: + - $a^{\mathrm{T}}b = 1 + 2 + 2 = 5$, $a^{\mathrm{T}}a = 3$ → $\hat{x} = \frac{5}{3}$. + - $p = (\frac{5}{3}, \frac{5}{3}, \frac{5}{3})$, $e = b - p = (-\frac{2}{3}, \frac{1}{3}, \frac{1}{3})$. + - 검산: $e^{\mathrm{T}}a = -\frac{2}{3} + \frac{1}{3} + \frac{1}{3} = 0$ ✓. + - (b) 계산: + - $a^{\mathrm{T}}b = -1 - 9 - 1 = -11$, $a^{\mathrm{T}}a = 11$ → $\hat{x} = -1$. + - $p = -a = (1, 3, 1) = b$! 오차 $e = 0$. + - 왜: $a = -b$ 라서 $b$ 가 이미 그 직선 위에 있다. [3단계] 특별한 경우 ①의 변형 (방향이 반대여도 같은 직선). + - 핵심 문장: $b$ 가 이미 직선 위에 있으면 사영은 자기 자신이고 $e = 0$ 이다. 부호 반대 방향도 같은 직선이다. + - **문제 8·9**: $b = (1,1)$ 을 $a_1 = (1,0)$, $a_2 = (1,2)$ 방향 직선에 각각 사영하고 합을 보라. 그리고 두 벡터가 이루는 평면 위로의 사영은? + - 📌 참고 그림 (p.231): + - ![그림 4.4와 4.5: 직교하는 기저와 직교하지 않는 기저 (p.231)](../assets/4장_p231_그림4.4_4.5_직교기저.png) + - 🔁 개념 상기: [2단계]에서 $p_1 + p_2 = b$ 가 성립한 건 두 공간이 **직교여공간**이었기 때문이다. 직교하지 않는 두 직선이면 어떻게 되는지 보는 문제. + - 1단계, 각각 사영: + - $a_1$ 위로: $\hat{x} = \frac{1}{1} = 1$ → $p_1 = (1, 0)$. + - $a_2$ 위로: $a_2^{\mathrm{T}}b = 3$, $a_2^{\mathrm{T}}a_2 = 5$ → $p_2 = \frac{3}{5}(1,2) = (\frac{3}{5}, \frac{6}{5})$. + - 2단계, 합해보기: $p_1 + p_2 = (\frac{8}{5}, \frac{6}{5}) \neq (1, 1) = b$. + - 이유: $a_1^{\mathrm{T}}a_2 = 1 \neq 0$, 두 직선이 직교하지 않아서 두 사영이 성분을 **중복으로** 집어간다 ([그림 4.5]의 상황). + - 대비 (문제 5~7, [그림 4.4]): 기저가 서로 직교하면 $P_1 + P_2 + P_3 = I$, 사영 합이 정확히 $b$ 가 된다. "직교 기저는 사영을 낱개로 쪼개 계산해도 된다"는 이 사실이 4.4절(정규직교기저)의 예고편이다. + - 3단계 (문제 9), 평면 위로 제대로 사영하면: $a_1, a_2$ 가 이루는 평면은 $\mathbb{R}^2$ 전체다. $A = \begin{bmatrix} 1&1 \\ 0&2 \end{bmatrix}$ 는 가역 정사각이므로 + - $P = A(A^{\mathrm{T}}A)^{-1}A^{\mathrm{T}} = AA^{-1}(A^{\mathrm{T}})^{-1}A^{\mathrm{T}} = I$ (가역일 땐 이 분리가 합법! [5단계] ⚠️의 뒤집기). + - 사영은 $p = Ib = b$, 오차 $e = 0$. 전체 공간 위로의 사영은 아무것도 바꾸지 않는다. + - 핵심 문장: 직교하지 않는 방향들로의 사영은 합쳐도 $b$ 가 안 된다. 낱개 사영의 합이 전체 사영이 되는 것은 직교 기저의 특권이다. + - **문제 14** (꼭 풀어볼 만한 문제): $A$ 는 $4 \times 4$ 항등행렬에서 마지막 열을 제거한 $4 \times 3$ 행렬이다. $b = (1,2,3,4)$ 를 $\mathbf{C}(A)$ 위로 사영하라. 사영행렬 $P$ 는? + - 🔁 개념 상기: 사영행렬 공식 $P = A(A^{\mathrm{T}}A)^{-1}A^{\mathrm{T}}$ ([5단계]). 계산 전에 열공간이 뭔지부터 보면 지름길이 보인다. + - 1단계, 열공간 파악: $A$ 의 열은 $e_1 = (1,0,0,0)$, $e_2 = (0,1,0,0)$, $e_3 = (0,0,1,0)$. 열공간은 "4번째 성분이 0인 벡터 전체" ($\mathbb{R}^4$ 안의 3차원 부분공간). + - 2단계, 공식 계산이 순식간에 끝난다: $A^{\mathrm{T}}A = I_3$ (열들이 서로 직교하고 크기 1이라서). 따라서 + - $$P = A(I_3)^{-1}A^{\mathrm{T}} = AA^{\mathrm{T}} = \begin{bmatrix} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&0 \end{bmatrix}$$ + - 3단계, 사영: $p = Pb = (1, 2, 3, 0)$. 4번째 성분만 지워진다 ([2단계]의 $xy$-평면 사영과 똑같은 원리, 한 차원 위 버전). + - 검산: $e = b - p = (0,0,0,4)$ 는 열공간의 세 기저 모두와 직교 ✓ ($e$ 는 4번째 축 방향). + - 💡 수위 조절: $A^{\mathrm{T}}A = I$ 가 되는 "정규직교 열"이면 사영행렬이 $P = AA^{\mathrm{T}}$ 로 초간단해진다. 이 좋은 성질을 일반 행렬에 만들어주는 것이 4.4절 그람-슈미트다. 지금은 이 맛보기만. + - 핵심 문장: 열들이 정규직교면 $A^{\mathrm{T}}A = I$ 라서 $P = AA^{\mathrm{T}}$, 사영은 "해당 성분만 남기기"가 된다. + - **문제 17**: 두 벡터 $(1,2,-1), (1,0,1)$ 의 선형결합 중 $b = (2,1,1)$ 과 가장 가까운 것은? + - 🔁 개념 상기: "가장 가까운 선형결합" = 열공간 위로의 사영 ([5단계]의 문제 설정 그대로). 그런데 공식에 넣기 전에 **$b$ 가 이미 그 평면 안인지** 확인하는 게 먼저다 (사영의 특별한 경우 ①). + - 1단계, $b$ 가 평면 안인지 검사: $c_1(1,2,-1) + c_2(1,0,1) = (2,1,1)$ 을 풀어본다. + - 2번째 성분: $2c_1 = 1$ → $c_1 = \frac{1}{2}$. + - 1번째 성분: $\frac{1}{2} + c_2 = 2$ → $c_2 = \frac{3}{2}$. + - 3번째 성분 검산: $-\frac{1}{2} + \frac{3}{2} = 1$ ✓ 일치! + - 2단계, 결론: $b$ 자체가 두 벡터의 선형결합이다. 가장 가까운 점은 $p = b = (2,1,1)$, 오차 $e = 0$, $Pb = b$. + - 핵심 문장: $b$ 가 이미 부분공간 안이면 사영은 $b$ 자신이다. 사영 문제는 "안인가 밖인가" 확인부터. + - **문제 18** (중요한 문제): $P^2 = P$ 이면 $(I-P)^2 = I-P$ 임을 보여라. $P$ 가 열공간 위로 사영할 때 $I - P$ 는 어디로 사영하는가? + - 🔁 개념 상기: $I - P$ 는 오차를 만드는 사영 ([4단계]). 사영행렬의 자격은 "두 번 해도 같음" ($P^2 = P$). + - 1단계, 전개 한 번이면 끝: + - $$(I-P)^2 = I - 2P + P^2 = I - 2P + P = I - P$$ + - 직관 번역: "오차의 오차"를 또 구하면 그대로다. $e$ 는 이미 여공간 위에 있으니까. + - 2단계, 빈칸: $P$ 가 $\mathbf{C}(A)$ 위로 사영하면 $I - P$ 는 그 직교여공간, 즉 **좌영공간** $\mathbf{N}(A^{\mathrm{T}})$ 위로 사영한다 ($e = b - p$ 가 사는 곳, [5단계] ❷). + - 확장 (문제 19의 두 사례로 감각 확인): + - $P$ 가 $\mathbb{R}^2$ 에서 $(1,1)$ 직선 위로 사영하면, $I - P$ 는 $(1,-1)$ 직선 위로 사영. + - $P$ 가 $\mathbb{R}^3$ 에서 $(1,1,1)$ 직선 위로 사영하면, $I - P$ 는 평면 $x + y + z = 0$ 위로 사영 (4.1 문제 17에서 구했던 바로 그 여공간 쌍!). + - 핵심 문장: $P$ 와 $I - P$ 는 직교여공간 쌍을 나눠 맡는 사영 콤비다 ($P + (I-P) = I$, $P(I-P) = 0$). + - **문제 20·21** (같은 평면, 두 가지 공법): 평면 $x - y - 2z = 0$ 위로의 사영행렬 $P$ 를 ① 평면의 기저로, ② 평면의 법선으로 각각 구하라. + - 🔁 개념 상기: ① 은 [5단계] 공식 (기저를 열로 → $P = A(A^{\mathrm{T}}A)^{-1}A^{\mathrm{T}}$), ② 는 [4단계] $I - P$ 아이디어 (여공간 쪽이 더 쉬우면 그쪽을 먼저 사영). 평면의 법선은 4.1의 언어로 $\mathbf{S}^{\perp}$ 의 기저다. + - 공법 ① (문제 20): 평면 위의 독립인 두 벡터를 고른다. 예: $(1,1,0)$ ($z=0$, $x=y$)과 $(2,0,1)$ ($y=0$, $x=2z$). + - $A = \begin{bmatrix} 1&2 \\ 1&0 \\ 0&1 \end{bmatrix}$, $A^{\mathrm{T}}A = \begin{bmatrix} 2&2 \\ 2&5 \end{bmatrix}$, $(A^{\mathrm{T}}A)^{-1} = \frac{1}{6}\begin{bmatrix} 5&-2 \\ -2&2 \end{bmatrix}$. + - $$P = A(A^{\mathrm{T}}A)^{-1}A^{\mathrm{T}} = \begin{bmatrix} 5/6&1/6&1/3 \\ 1/6&5/6&-1/3 \\ 1/3&-1/3&1/3 \end{bmatrix}$$ + - 공법 ② (문제 21): 평면의 방정식 $x - y - 2z = 0$ 에서 법선 $e = (1, -1, -2)$ 를 바로 읽는다 (3장: 평면 방정식의 계수 = 법선). + - 법선 직선 위로의 사영행렬 (랭크 1 공식): $Q = \frac{ee^{\mathrm{T}}}{e^{\mathrm{T}}e} = \frac{1}{6}\begin{bmatrix} 1&-1&-2 \\ -1&1&2 \\ -2&2&4 \end{bmatrix}$. + - 평면은 법선의 직교여공간이므로: $P = I - Q = \begin{bmatrix} 5/6&1/6&1/3 \\ 1/6&5/6&-1/3 \\ 1/3&-1/3&1/3 \end{bmatrix}$. 공법 ①과 정확히 일치 ✓. + - 검산: $P e = 0$ 확인 (법선은 평면 사영에서 죽어야 함): 1행 $\cdot e = \frac{5}{6} - \frac{1}{6} - \frac{2}{3} = 0$ ✓. + - 💡 실전 꿀팁: $\mathbb{R}^3$ 의 평면 사영은 공법 ②가 훨씬 빠르다 (기저 2개짜리 역행렬 대신 법선 1개짜리 나누기). 차원이 낮은 쪽($\mathbf{S}$ 든 $\mathbf{S}^{\perp}$ 든)을 사영하고 필요하면 $I$ 에서 빼라. + - 핵심 문장: 평면 사영행렬 = $I - (\text{법선 사영행렬})$. 여공간 쌍은 사영행렬도 $I$ 를 나눠 갖는다. diff --git a/Major Study/pages/4장 직교성.md b/Major Study/pages/4장 직교성.md index c2f9914..e8c5767 100755 --- a/Major Study/pages/4장 직교성.md +++ b/Major Study/pages/4장 직교성.md @@ -15,3 +15,4 @@ deck:: Logseq/전공공부/선형대수학 - 💡 수위 조절: 서론(p.208~209)은 사영·최소제곱·$QR$ 을 예고편처럼 쏟아붓는다. 여기서 다 이해될 필요가 전혀 없다. 각 절에서 제대로 배우니, 지금은 "직교가 이 모든 것의 공통 무기"라는 감만 잡으면 충분하다. - ## 소단원 - [[4.1 벡터와 부분공간의 직교성]] + - [[4.2 직선과 부분공간 위로의 사영]] \ No newline at end of file diff --git a/Major Study/pages/hls__4장_직교성_1786280501491_0.md b/Major Study/pages/hls__4장_직교성_1786280501491_0.md index 80a3ff3..5343671 100755 --- a/Major Study/pages/hls__4장_직교성_1786280501491_0.md +++ b/Major Study/pages/hls__4장_직교성_1786280501491_0.md @@ -112,4 +112,138 @@ file-path:: ../assets/4장_직교성_1786280501491_0.pdf ls-type:: annotation hl-page:: 4 hl-color:: yellow - id:: 6a7888b9-0f6a-494b-a9e7-3cf7dbe5dd91 \ No newline at end of file + id:: 6a7888b9-0f6a-494b-a9e7-3cf7dbe5dd91 +- 벡터 b는 S 안에서 가장 가까운 점 p로 사영된다. 그러면 오차벡터 e = b - p는 부분공간 S와 직교한다. + ls-type:: annotation + hl-page:: 13 + hl-color:: yellow + id:: 6a7a03f4-57ff-42c6-9820-cc0446588b7a +- 행렬 A가 선형독립인 열을 갖는다면, 벡터 b의 C(A) 위로의 사영은 p = A(AᵀA)⁻¹Aᵀb이다. 여기서 사영행렬 P = A(AᵀA)⁻¹Aᵀ는 대칭행렬이다 + ls-type:: annotation + hl-page:: 13 + hl-color:: yellow + id:: 6a7a0403-2799-49a5-952d-9fdf5d50e671 +- P² = P이다. 즉 두 번째 사영은 p를 p로 사영하므로 아무것도 변하지 않는다 + ls-type:: annotation + hl-page:: 13 + hl-color:: yellow + id:: 6a7a040f-bcd4-4e55-863a-6152ed91aa7f +- 벡터 b가 직선 위로 사영되면, 사영 p는 벡터 b의 해당 직선 방향 부분이 된다. 벡터 b가 평면 위로 사영되면, 사영 p는 벡터 b의 해당 평면 방향 부분이 된다. 사영 p는 Pb이다. + ls-type:: annotation + hl-page:: 13 + hl-color:: yellow + id:: 6a7a0415-779e-42a1-9b91-95340aa2264b +- [:span] + ls-type:: annotation + hl-page:: 14 + hl-color:: yellow + id:: 6a7a041f-7af3-4273-a7e3-c56d9b591165 + hl-type:: area + hl-stamp:: 1786381341457 +- 직선과 평면은 단순히 직교하는 것을 넘어, 서로 직교여공간이다 + ls-type:: annotation + hl-page:: 14 + hl-color:: yellow + id:: 6a7a042f-2b0c-4ecd-949e-f28c59939ff2 +- 벡터들은 p₁ + p₂ = b를 만족하고, 행렬들은 P₁ + P₂ = I, P₁P₂ = 0을 만족한다. + ls-type:: annotation + hl-page:: 14 + hl-color:: yellow + id:: 6a7a0437-3040-4bc0-a638-f5b5d33a8f99 +- b에서 p까지의 직선은 벡터 a와 직교한다 + ls-type:: annotation + hl-page:: 15 + hl-color:: yellow + id:: 6a7a043f-522f-490b-b277-c1d2a36f849c +- [:span] + ls-type:: annotation + hl-page:: 16 + hl-color:: yellow + id:: 6a7a044a-2260-4606-b890-6f4f9333742c + hl-type:: area + hl-stamp:: 1786381386523 +- [:span] + ls-type:: annotation + hl-page:: 16 + hl-color:: yellow + id:: 6a7a045f-fe09-47a3-b9b0-74e54a3248ee + hl-type:: area + hl-stamp:: 1786381407053 +- [:span] + ls-type:: annotation + hl-page:: 17 + hl-color:: yellow + id:: 6a7a046a-3e0f-494b-b603-5a93cc3eeed4 + hl-type:: area + hl-stamp:: 1786381417724 +- 항등행렬 I에 대해 행렬 I - P도 사영행렬이어야 한다 + ls-type:: annotation + hl-page:: 17 + hl-color:: yellow + id:: 6a7af20e-41d5-4040-a603-a53e5809a41f +- P가 하나의 부분공간 위로 사영할 때, I - P는 그 부분공간과 직교하는 부분공간 위로 사영한다. + ls-type:: annotation + hl-page:: 17 + hl-color:: yellow + id:: 6a7af214-6b8f-4d35-99ab-549c781aa104 +- p = Ax̂ (사영) + ls-type:: annotation + hl-page:: 18 + hl-color:: yellow + id:: 6a7af225-e248-4100-b418-ce9559de344b +- x̂ 위의 ^(hat) 표기는 열공간 안에서 가장 가까운 벡터를 제공하는 최적의 선택임을 의미한다. n = 1인 경우, 그 최적의 선택은 x̂ = aᵀb/aᵀa이다 + ls-type:: annotation + hl-page:: 18 + hl-color:: yellow + id:: 6a7af22c-331d-4718-b45f-3c7997457022 +- (S) 안에서 벡터 x̂을 구하고, (C) 안에서 사영 p = Ax̂을 구한 다음, (T) 안에서 사영행렬 P를 구해보자. + ls-type:: annotation + hl-page:: 18 + hl-color:: yellow + id:: 6a7af232-2329-4ae5-8b0b-f32445874c99 +- 오차벡터 b - Ax̂은 부분공간과 직교한다 + ls-type:: annotation + hl-page:: 18 + hl-color:: yellow + id:: 6a7af237-5421-4b5e-9aaf-f18ab9f7b198 +- [:span] + ls-type:: annotation + hl-page:: 19 + hl-color:: yellow + id:: 6a7af245-203a-4d34-99d3-63a1f7a3bc3e + hl-type:: area + hl-stamp:: 1786442309180 +- 핵심은 정규방정식(normal equation) Aᵀ(b - Ax̂) = 0을 해결하는 단계이다. + ls-type:: annotation + hl-page:: 19 + hl-color:: yellow + id:: 6a7af250-44f8-4e4d-8c66-e774cd33390a +- [:span] + ls-type:: annotation + hl-page:: 19 + hl-color:: yellow + id:: 6a7af25d-f986-4721-a903-e7463d07eb9a + hl-type:: area + hl-stamp:: 1786442333608 +- 행렬 P = A(AᵀA)⁻¹Aᵀ를 계산할 때 주의할 점이 있다 + ls-type:: annotation + hl-page:: 20 + hl-color:: yellow + id:: 6a7af267-9dca-45c0-bc40-f0a8db37093b +- A는 직사각행렬이다. 즉 A의 역행렬은 없다. m > n인 경우에는 A⁻¹가 존재하지 않으므로 (AᵀA)⁻¹를 A⁻¹와 (Aᵀ)⁻¹로 나눌 수 없다 + ls-type:: annotation + hl-page:: 20 + hl-color:: yellow + id:: 6a7af26d-7da7-41db-ba30-fac9454a16e0 +- AᵀA가 가역행렬일 필요충분조건은 행렬 A의 열이 선형독립인 것이다. + ls-type:: annotation + hl-page:: 21 + hl-color:: yellow + id:: 6a7af283-66b3-4560-ae49-96b3acabb433 +- [:span] + ls-type:: annotation + hl-page:: 22 + hl-color:: yellow + id:: 6a7af28d-1121-4a8f-94c5-3749f4744e14 + hl-type:: area + hl-stamp:: 1786442380971 \ No newline at end of file

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