From f2af66dc537d042b07567738b14cda8eed91eb53 Mon Sep 17 00:00:00 2001 From: songyc macbook Date: Wed, 19 Aug 2026 17:11:22 +0900 Subject: [PATCH] =?UTF-8?q?4.5=20=ED=96=89=EB=A0=AC=EC=9D=98=20=EC=9C=A0?= =?UTF-8?q?=EC=82=AC=EC=97=AD=ED=96=89=EB=A0=AC:=20=EC=86=8C=EB=8B=A8?= =?UTF-8?q?=EC=9B=90=20=EC=A0=95=EB=A6=AC=20(=EB=A9=94=EB=AA=A8=208?= =?UTF-8?q?=EA=B1=B4=20=EB=B0=98=EC=98=81,=20=EC=B2=98=EC=9D=8C=EB=B6=80?= =?UTF-8?q?=ED=84=B0=20=EC=84=A4=EB=AA=85=20=EB=AA=A8=EB=93=9C)=20+=204?= =?UTF-8?q?=EC=9E=A5=20=EA=B3=A0=EC=B0=B0=C2=B7=ED=95=B5=EC=8B=AC=EB=AC=B8?= =?UTF-8?q?=EC=A0=9C=20=EC=9E=91=EC=84=B1=20(4=EC=9E=A5=20=EC=99=84?= =?UTF-8?q?=EA=B2=B0)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit - 8단계 재구성: 3.3 해 유형표 → 좌역=최소제곱·우역=최단 해 연결, AᵀA/AAᵀ 모순 해소, A⁺ 정의·그림 4.11·사영 정체·계산법 3종 - 하이라이트 19개 연결 (텍스트 7 + 영역 12), OCR 교정 7건, 이미지 13개 추출 - 연습문제 1·2·4·6·11 선정·해설, 독립 검증 워크플로 통과 (지적 2건 수정) - 4장 고찰: 책 ❶~❾ 정리 + 장 전체 한 줄 + 절별 회고 + 자가 질문 목록 - 이 단원 핵심문제 5선 (4.1-3, 4.2-20·21, 4.3-1, 4.4-15, 4.5-1, 블록 링크) Co-Authored-By: Claude Fable 5 Claude-Session: https://claude.ai/code/session_01Ufr6wUhJAzBaqcaBiRJ1xk --- .../assets/4장_p268_역행렬_3형제_도입표.png | Bin 0 -> 499791 bytes .../assets/4장_p269_좌역_우역_비교표.png | Bin 0 -> 589840 bytes .../4장_p269_최소노름_최소제곱해_박스.png | Bin 0 -> 105016 bytes .../assets/4장_p270_A플러스_정의_박스.png | Bin 0 -> 82635 bytes .../4장_p270_그림4.11_A플러스_큰그림.png | Bin 0 -> 610989 bytes .../assets/4장_p271_Prow_Pcolumn_식.png | Bin 0 -> 117157 bytes Major Study/assets/4장_p273_CR_구성_박스.png | Bin 0 -> 238951 bytes .../assets/4장_p273_식3_C플러스_R플러스.png | Bin 0 -> 67194 bytes .../assets/4장_p273_식4_유사역행렬.png | Bin 0 -> 169542 bytes .../assets/4장_p273_식5_펜로즈_조건.png | Bin 0 -> 112547 bytes Major Study/assets/4장_p275_가역변환_박스.png | Bin 0 -> 238035 bytes Major Study/assets/4장_p275_식7_CR_유사역.png | Bin 0 -> 68901 bytes Major Study/assets/4장_p275_식8_사영_확인.png | Bin 0 -> 119868 bytes .../assets/4장_직교성_1786280501491_0.edn | 277 +++++++++++++++++- 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111.77301025390625, + :width 716, + :height 1012.6281590388325}, + :rects ({:x1 168.6089324951172, + :y1 71.23480224609375, + :x2 618.9344482421875, + :y2 85.12368774414062, + :width 716, + :height 1012.6281590388325} + {:x1 70.73567199707031, + :y1 97.88412475585938, + :x2 214.11473083496094, + :y2 111.77301025390625, + :width 716, + :height 1012.6281590388325}), + :page 63}, + :content {:text "모든 행렬 A는 유사역행렬 A +를 갖는다. 유사 역행렬은 역행렬 , 좌역행렬 , 우역행렬을 모두 포함 한다."}, + :properties {:color "yellow"}} + {:id #uuid "6a855f7a-997f-4ca8-95fb-d572477bd8e8", + :page 63, + :position {:bounding {:x1 59, + :y1 115.6640625, + :x2 630, + :y2 379.6640625, + :width 716, + :height 1012.6281590388325}, + :rects (), + :page 63}, + :content {:text "[:span]", :image 1787125625873}, + :properties {:color "yellow"}} + {:id #uuid "6a855fcf-ec62-4b22-a9ec-e7d5d880455e", + :page 63, + :position {:bounding {:x1 71.66883087158203, + :y1 624.3381118774414, + :x2 486.26300048828125, + :y2 638.9214248657227, + :width 716, 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:content {:text "2가 영 공간 성분으로 0 을 갖는다"}, + :properties {:color "yellow"}} + {:id #uuid "6a855ff4-89f2-4e1d-a036-be210e4b63ab", + :page 64, + :position {:bounding {:x1 82, + :y1 468.5234375, + :x2 666, + :y2 826.5234375, + :width 716, + :height 1012.6281590388325}, + :rects (), + :page 64}, + :content {:text "[:span]", :image 1787125748462}, + :properties {:color "yellow"}} + {:id #uuid "6a856002-40c1-44eb-acc5-594e6e5f8c2a", + :page 64, + :position {:bounding {:x1 110, + :y1 255.71875, + :x2 643, + :y2 295.71875, + :width 716, + :height 1012.6281590388325}, + :rects (), + :page 64}, + :content {:text "[:span]", :image 1787125762067}, + :properties {:color "yellow"}} + {:id #uuid "6a85600c-3cac-4ba4-9f76-bc8e55a4ff1c", + :page 65, + :position {:bounding {:x1 57, + :y1 137.6640625, + :x2 633, + :y2 184.6640625, + :width 716, + :height 1012.6281590388325}, + :rects (), + :page 65}, + :content {:text "[:span]", :image 1787125772423}, + :properties {:color "yellow"}} + {:id #uuid "6a85601e-b15a-432c-a104-43aa309f4d34", + :page 66, + :position {:bounding {:x1 133.4526824951172, + :y1 705.3710556030273, + :x2 623.96337890625, + :y2 719.2599716186523, + :width 716, + :height 1012.6281590388325}, + :rects ({:x1 133.4526824951172, + :y1 705.3710556030273, + :x2 623.96337890625, + :y2 719.2599716186523, + :width 716, + :height 1012.6281590388325}), + :page 66}, + :content {:text "행공간 의 두 벡터 z와 X가 열공간의 동일한 벡터 Az= AX 로 갈 수 는 없다는 것"}, + :properties {:color "yellow"}} + {:id #uuid "6a85602c-bc22-4eb5-86c2-1079716484c5", + :page 66, + :position {:bounding {:x1 115.62499237060547, + :y1 735.0585556030273, + :x2 653.9713134765625, + :y2 748.9474716186523, + :width 716, + :height 1012.6281590388325}, + :rects ({:x1 115.62499237060547, + :y1 735.0585556030273, + :x2 653.9713134765625, + :y2 748.9474716186523, + :width 716, + :height 1012.6281590388325}), + :page 66}, + :content {:text "열공간의 모든 벡터 b= Az는 행 공간 의 단 하나 의 벡터 z 로부터 나온다 . 그러면 A+ b =이다 ."}, + :properties {:color "yellow"}} + {:id #uuid "6a856034-75b4-4025-8463-6bbce9302041", + :page 67, + :position {:bounding {:x1 53, + :y1 209.5234375, + :x2 624, + :y2 311.5234375, + :width 716, + :height 1012.6281590388325}, + :rects (), + :page 67}, + :content {:text "[:span]", :image 1787125811831}, + :properties {:color "yellow"}} + {:id #uuid "6a85603b-9d37-4456-b37c-0732cb7b80d3", + :page 67, + :position {:bounding {:x1 68, + :y1 458.96875, + :x2 622, + :y2 500.96875, + :width 716, + :height 1012.6281590388325}, + :rects (), + :page 67}, + :content {:text "[:span]", :image 1787125818753}, + :properties {:color "yellow"}} + {:id #uuid "6a856040-be47-4a73-a52b-d486b76c8d2f", + :page 67, + :position {:bounding {:x1 59, + :y1 567.96875, + :x2 619, + :y2 646.96875, + :width 716, + :height 1012.6281590388325}, + :rects (), + :page 67}, + :content {:text "[:span]", :image 1787125824274}, + :properties {:color "yellow"}} + {:id #uuid "6a856047-b418-487d-873c-2967c2407488", + :page 67, + :position {:bounding 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{:color "yellow"}} + {:id #uuid "6a856077-68be-4fbd-b9af-716de18b6602", + :page 69, + :position {:bounding {:x1 67, + :y1 504.25, + :x2 618, + :y2 544.25, + :width 716, + :height 1012.6281590388325}, + :rects (), + :page 69}, + :content {:text "[:span]", :image 1787125878829}, + :properties {:color "yellow"}} + {:id #uuid "6a856080-57ec-455d-8096-8c361f5939fc", + :page 69, + :position {:bounding {:x1 65.28862762451172, + :y1 553.1249694824219, + :x2 613.1637573242188, + :y2 593.5546569824219, + :width 716, + :height 1012.6281590388325}, + :rects ({:x1 65.28862762451172, + :y1 553.1249694824219, + :x2 613.1637573242188, + :y2 567.7082824707031, + :width 716, + :height 1012.6281590388325} + {:x1 66.29773712158203, + :y1 579.6658020019531, + :x2 241.80441284179688, + :y2 593.5546569824219, + :width 716, + :height 1012.6281590388325}), + :page 69}, + :content {:text "AT와 AT는 동일한 행공간 과 공간을 갖는다 . 둘 의 차이점은 A + A가 행공간 위로 사영하는대칭인 사영행렬이라는 것이다"}, + :properties {:color "yellow"}} + {:id #uuid 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zsx|PAn0~!a3fbvIQqhx&%_Habx5HxO1`2gU3`JCXfv07BY7)szxb73`1=#UQ})VL-sA+ZKkX^@sUL} zmwo|$MIF1TcIy3JR))1UWlRe~+mDe3+(HL6+?rmjEY>)mJ|J^O^peP9`o<#%j(-T6 zQ7C^R$8b;-I1uFS)x`4PgI;3qKo+YhNZ&M`gAF2}oCQy)ZW1z~hcBCvGRX|S7se09H zg+FG>xqC`SI5>Idv8(HP&c8o~gj0Ypd`T-v6Nn(8hL5J8e+?pYzWKtJd_ z)y7AgGtqG7m@~yIP_(O_$kfhs>@RTQPKp{4xDzLP8=$U))=lxh?oB1fml3D6d{M-|UeFrn!Gg9+nN@xuWP z@stbJ@zvgVrgEElFt|vT4v_{l9bLG<-Dv9K=wIOY2K5nauCxMp&q~v%&9Az;4q8z{ ztz^Q$NS*jJ)OmoDYK;Gh01mvvkx n$, 열 독립, $r = n$). 방정식이 미지수보다 많아 **해가 없을 수 있는** 상황. + - 이름의 뜻: **왼쪽에서 곱했을 때만** $I$ 가 되는 행렬. $A^{+}A = I$ ($n \times n$). 오른쪽 곱 $AA^{+}$ 는 $I$ 가 아니다. + - 공식은 새로 외울 것이 없다. 이미 아는 재료의 조립이다: + - 열 독립 → $A^{\mathrm{T}}A$ ($n \times n$) 가역 (4.2 [6단계]의 그 정리!) → + - $$A^{+} = (A^{\mathrm{T}}A)^{-1}A^{\mathrm{T}} \qquad \text{검산: } A^{+}A = (A^{\mathrm{T}}A)^{-1}(A^{\mathrm{T}}A) = I \ ✓$$ + - ✅ **"이것이 바로 최소제곱법"이 무슨 소리냐면**: 이 $A^{+}$ 를 $b$ 에 곱해보라. + - $$A^{+}b = (A^{\mathrm{T}}A)^{-1}A^{\mathrm{T}}b = \hat{x}$$ + - 우변은 **4.3 정규방정식의 해 그 자체**다! 즉 좌역행렬은 "최소제곱해 계산기"를 행렬 하나로 포장한 것. $A^{+}b$ 를 계산하는 행위 = 최소제곱법을 실행하는 행위. 그래서 책이 그 문장을 쓴 것이다. + - $b$ 가 열공간 안이면 정확한 해를, 밖이면 최선의 해를 돌려준다 (4.3에서 확인한 "최소제곱은 상위호환" 그대로). + - 그럼 반대쪽 곱 $AA^{+}$ 는 뭐가 되나: $A(A^{\mathrm{T}}A)^{-1}A^{\mathrm{T}}$, 어디서 많이 봤다. **4.2의 사영행렬** $P$ **그 자체!** 왼쪽 곱은 $I$ (계수 세계에서는 완벽), 오른쪽 곱은 열공간 사영 (벡터 세계에서는 "갈 수 있는 데까지만"). + - 🔢 제일 작은 예: $A = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$ (세로, $r = n = 1$). + - $A^{\mathrm{T}}A = 2$ → $A^{+} = \frac{1}{2}\begin{bmatrix} 1 & 1 \end{bmatrix}$. + - $A^{+}A = \frac{1}{2}(1 + 1) = 1 = I_1$ ✓. 반대쪽 $AA^{+} = \frac{1}{2}\begin{bmatrix} 1&1 \\ 1&1 \end{bmatrix}$ = 상수벡터 방향 사영행렬 ($\neq I$). + - $A^{+}b = \frac{b_1 + b_2}{2}$ = **평균**! (4.3 문제 12의 "평균 = 사영"이 좌역행렬의 언어로 재등장. 세로 $A = (1,1)$ 의 좌역은 평균 기계다.) + - #### [4단계] 우역행렬: "최소길이 해"의 뜻 + - 대상: **가로로 긴** $A$ ($m < n$, 행 독립, $r = m$). 미지수가 방정식보다 많아 **해가 무한한** 상황. + - 이름의 뜻: **오른쪽에서 곱했을 때만** $I$: $AA^{+} = I$ ($m \times m$). 왼쪽 곱 $A^{+}A$ 는 $I$ 가 아니다. + - 공식도 좌역과 거울 대칭: 행 독립 → $AA^{\mathrm{T}}$ ($m \times m$) 가역 → + - $$A^{+} = A^{\mathrm{T}}(AA^{\mathrm{T}})^{-1} \qquad \text{검산: } AA^{+} = (AA^{\mathrm{T}})(AA^{\mathrm{T}})^{-1} = I \ ✓$$ + - ✅ **"최소길이 해(최단거리 해)"가 무슨 소리냐면**: 해가 무한하니 고르는 문제가 남는다. + - 3.3의 완전해: $x = x_p + x_n$ (특수해 + 영공간 아무거나). 전부 같은 $b$ 로 가는 정당한 해들. + - 그중 $x^{+} = A^{+}b$ 는 **행공간에 들어 있는 유일한 해**다. 즉 영공간 성분이 0. + - 왜 그게 최단인가: 4.1의 직교 분해로 임의의 해는 $x = x^{+} + x_n$ (행공간 성분 ⊥ 영공간 성분) → 피타고라스: + - $$\|x\|^2 = \|x^{+}\|^2 + \|x_n\|^2 \geq \|x^{+}\|^2$$ + - 영공간 성분은 **길이만 늘리는 군살**이다. 그걸 0으로 만든 해가 최단. 4.3 [7단계]의 종속 열 예제에서 "최단 해 $x^{+} = (1,1)$"이라 했던 그 말의 정체가 이것이다. + - 🔢 제일 작은 예: $A = \begin{bmatrix} 1 & 1 \end{bmatrix}$ (가로, $r = m = 1$). + - $AA^{\mathrm{T}} = 2$ → $A^{+} = \frac{1}{2}\begin{bmatrix} 1 \\ 1 \end{bmatrix}$. $AA^{+} = \frac{1}{2}(1+1) = 1$ ✓. + - $Ax = b$ 는 $x_1 + x_2 = b$: 해가 직선 하나 가득. $A^{+}b = (\frac{b}{2}, \frac{b}{2})$ 는 그 직선 위에서 **원점에 가장 가까운 점** (수선의 발!). + - 검산: 다른 해 $(b, 0)$ 의 길이는 $|b|$, 최단 해의 길이는 $\frac{|b|}{\sqrt{2}}$. 정말 더 짧다. + - ⚠️ 오해 교정 ("좌역·우역행렬은 $I$ 에 0 붙인 모양?"): 아니다. 좌역·우역행렬 **자신**은 $\frac{1}{2}[1\ 1]$ 처럼 평범한 직사각행렬이다. "$I$ 랭크만큼 + 나머지 0" 모양이 나오는 곳은 두 군데 뿐이다: ① [1단계] 도입 표의 특수 예 ($I$ 를 잘라 만든 $I_L, I_R$), ② [7단계]에서 볼 $A^{+}A$, $AA^{+}$ (사영행렬을 좋은 기저에서 본 모습). 그 인상이 섞인 것. + - #### [5단계] $A^{\mathrm{T}}A$ 대 $AA^{\mathrm{T}}$: 모순이 아니다 (기억 교정) + - 팩트 체크: **둘 다 언제나 정사각이다.** $A$ 가 $m \times n$ 이면 $A^{\mathrm{T}}A$ 는 $n \times n$, $AA^{\mathrm{T}}$ 는 $m \times m$ (4.2 스스로 점검 "내적 표" 문답 그대로: 열 내적 표는 열 개수만큼, 행 내적 표는 행 개수만큼). + - "$AA^{\mathrm{T}}$ 는 정사각이 아니라고 하지 않았나?"는 기억의 혼선이다. 아마 4.4의 "**직사각** $Q$ **에서** $QQ^{\mathrm{T}} \neq I$" 와 섞였을 것. **항등이 아닌 것**과 **정사각이 아닌 것**은 다른 말이다. $QQ^{\mathrm{T}}$ 도 어엿한 $m \times m$ 정사각(사영행렬)이었다. + - 둘의 진짜 차이는 크기가 아니라 **가역성 조건**이다: + - | 행렬 | 크기 | 가역 ⟺ | 어느 역행렬의 재료? | + | --- | --- | --- | --- | + | $A^{\mathrm{T}}A$ | $n \times n$ | 열 독립 ($r = n$) | 좌역 $(A^{\mathrm{T}}A)^{-1}A^{\mathrm{T}}$ | + | $AA^{\mathrm{T}}$ | $m \times m$ | 행 독립 ($r = m$) | 우역 $A^{\mathrm{T}}(AA^{\mathrm{T}})^{-1}$ | + - 세로(열 독립)는 $A^{\mathrm{T}}A$ 가 살아 있어 좌역을, 가로(행 독립)는 $AA^{\mathrm{T}}$ 가 살아 있어 우역을 만든다. 이 대응이 [2단계] 표와 맞물린다. + - #### [6단계] 진짜 유사역행렬 $A^{+}$: 랭크가 둘 다 부족해도 된다 + - 이제 최악의 상황: $r < m$ 이고 $r < n$ (해가 0개 아니면 ∞개, 한쪽 역행렬도 없음). 여기서도 작동하는 만능 되돌리기를 정의한다. + - **정의는 행동강령 3줄이다** (책 ❶❷❸): $\mathbb{R}^m$ 의 임의 벡터 $y$ 를 4.1처럼 직교 분해한다: $y = b + z$ ($b \in$ 열공간, $z \in$ 좌영공간). + - ① 열공간 성분 $b$: 그 유일한 조상에게 되돌린다. $b = Ax$ 인 **행공간의 유일한** $x$ 로: $A^{+}b = x$. + - ② 좌영공간 성분 $z$: 되돌릴 곳이 없으니 죽인다: $A^{+}z = 0$. + - ③ 합치면: $A^{+}y = A^{+}b + A^{+}z = x$. + - 📌 원문 정의 박스 (p.270): [🔗 PDF 위치](((6a856002-40c1-44eb-acc5-594e6e5f8c2a))) + - ![A⁺의 정의 (p.270)](../assets/4장_p270_A플러스_정의_박스.png) + - 이 정의가 가능한 근거 = **4.1 [5단계]의 그 정리**: "$A$ 는 행공간에서 열공간으로의 가역변환이다". + - 📌 원문 (p.272, 275): ((6a85601e-b15a-432c-a104-43aa309f4d34)) / ((6a85602c-bc22-4eb5-86c2-1079716484c5)) / ((6a856065-6171-48c3-88a1-51dd03f8ea10)) + - 증명도 4.1에서 했던 그대로: 행공간의 $x \neq X$ 가 $Ax = AX$ 라면 $x - X$ 가 행공간과 영공간에 동시에 있어 0. 모순. + - $A^{+}$ 는 그 가역변환의 **역** + "복원 불가능한 부분(좌영공간)은 0으로" 규칙을 합친 것. **가능한 경우** $A^{+}$ **는** $A$ **를 역변환한다** (열공간 → 행공간). + - 📌 원문 가역변환 박스 (p.275): [🔗 PDF 위치](((6a85606a-40c4-4d49-be45-8b3997d0b687))) + - ![행공간↔열공간 가역변환 (p.275)](../assets/4장_p275_가역변환_박스.png) + - **[그림 4.11] = 그림 4.1의 화살표를 전부 거꾸로 돌린 그림**: + - 📌 원문 그림 4.11 (p.270): [🔗 PDF 위치](((6a855ff4-89f2-4e1d-a036-be210e4b63ab))) + - ![그림 4.11 A⁺의 큰 그림 (p.270)](../assets/4장_p270_그림4.11_A플러스_큰그림.png) + - 읽는 법: $A$ 는 왼쪽(행공간) → 오른쪽(열공간)으로, $A^{+}$ 는 오른쪽 → 왼쪽으로. $A^{+}$ 의 4가지 기본 부분공간은 $A^{\mathrm{T}}$ 와 동일 (행공간↔열공간이 서로 바뀜, 좌영공간은 $A^{+}$ 의 영공간). + - 그래서 일반 $b$ 에 대한 $x^{+} = A^{+}b$ 의 신분: + - 📌 원문 (p.269): [🔗 PDF 위치](((6a855fda-fea7-4d81-972e-04718e48d53c))) / ((6a855fcf-ec62-4b22-a9ec-e7d5d880455e)) / ((6a855fe0-2eb2-4a14-91bf-9a54a70ea013)) + - ![x⁺는 최소노름 최소제곱해 (p.269)](../assets/4장_p269_최소노름_최소제곱해_박스.png) + - $x^{+}$ 는 **최소노름 최소제곱해**: 오차 $\|Ax - b\|^2$ 을 최소화하는 후보들([3단계] 처방) 중에서 **길이** $\|x\|$ **가 가장 짧은 것**([4단계] 처방). 두 처방의 동시 적용. + - 핵심 성질: $x^{+}$ 는 영공간 성분이 0 (행공간 소속). 4.3 종속 열 예제의 $x^{+} = (1, 1)$ 이 드디어 정식으로 정의된 것. + - #### [7단계] $A^{+}A$ 와 $AA^{+}$ 의 정체: $I$ 가 못 되면 사영이 된다 + - 진짜 역행렬이면 $A^{-1}A = AA^{-1} = I$. 유사역행렬은 거기까지는 못 가고, **갈 수 있는 최선**에 도달한다: + - 📌 원문 (p.271, 275): [🔗 PDF 위치](((6a85600c-3cac-4ba4-9f76-bc8e55a4ff1c))) / ((6a856080-57ec-455d-8096-8c361f5939fc)) + - ![P_row와 P_column (p.271)](../assets/4장_p271_Prow_Pcolumn_식.png) + - $$A^{+}A = P_{\text{row}}\ (\text{행공간 사영, } n \times n) \qquad AA^{+} = P_{\text{column}}\ (\text{열공간 사영, } m \times m)$$ + - 왜 사영인가 (한 줄 추적): $A^{+}Ax$ 에서 $Ax$ 는 $x$ 의 **행공간 성분만** 반영하고 ($Ax_n = 0$), $A^{+}$ 는 그 결과를 행공간으로 되돌린다. 그러니 $x \to (x$ 의 행공간 성분$)$: 정확히 행공간 사영. + - 좋은 기저(행공간 기저 $r$ 개 + 영공간 기저 $n - r$ 개)로 보면 이 사영행렬이 $\begin{bmatrix} I_r & 0 \\ 0 & 0 \end{bmatrix}$ 모양이 된다 (그림 4.11 의 라벨). "랭크만큼 $I$, 나머지 0"이라는 직감이 여기서는 정확히 맞는다. + - 특수화 확인: 좌역 상황($r = n$)이면 행공간 = $\mathbb{R}^n$ 전체 → $P_{\text{row}} = I$ → $A^{+}A = I$ ✓. 우역 상황($r = m$)이면 열공간 = $\mathbb{R}^m$ 전체 → $AA^{+} = I$ ✓. [3]·[4단계]가 이 그림의 특수 케이스였다. + - #### [8단계] 계산법 3종과 펜로즈 조건 (수위 조절: 눈도장 수준이면 충분) + - **① 대각행렬** $D$: 제일 쉬운 경우. 대각 성분의 역수를 취하되 $\frac{1}{0}$ **은 0으로** 두고, 모양은 전치. + - $$D = \begin{bmatrix} 2&0&0 \\ 0&3&0 \\ 0&0&0 \end{bmatrix} \Rightarrow D^{+} = \begin{bmatrix} 1/2&0&0 \\ 0&1/3&0 \\ 0&0&0 \end{bmatrix}$$ + - "역수 없는 자리(0)는 포기하고 0"이 유사역의 정신을 그대로 보여준다 (복원 불가능한 방향은 0으로). + - **② 한쪽 역이 있는 경우**: [3]·[4단계] 공식. 세로 → $(A^{\mathrm{T}}A)^{-1}A^{\mathrm{T}}$, 가로 → $A^{\mathrm{T}}(AA^{\mathrm{T}})^{-1}$. + - **③ 일반 (랭크 부족)**: $A = CR$ (3장의 그 분해!)로 쪼갠 뒤, $C$ 는 열 독립이니 좌역을, $R$ 는 행 독립이니 우역을 갖는다. 운 좋게 그 둘을 이어 붙이면 끝: + - 📌 원문 (p.273): [🔗 PDF 위치](((6a856034-75b4-4025-8463-6bbce9302041))) / 식 (3) [🔗](((6a85603b-9d37-4456-b37c-0732cb7b80d3))) / 식 (4) [🔗](((6a856040-be47-4a73-a52b-d486b76c8d2f))) + - $$A^{+} = R^{+}C^{+} = R^{\mathrm{T}}(C^{\mathrm{T}}AR^{\mathrm{T}})^{-1}C^{\mathrm{T}} \quad (\text{식 (7)})$$ + - 📌 원문 식 (7)·(8) (p.275): [🔗 PDF 위치](((6a856077-68be-4fbd-b9af-716de18b6602))) / [🔗 PDF 위치](((6a856088-3271-41fd-b568-8a0c1b029b59))) + - 지금은 "이런 공식이 있다" 눈도장이면 충분하다. 진짜 계산의 왕도는 7장의 SVD ($A = U\Sigma V^{\mathrm{T}} \to A^{+} = V\Sigma^{+}U^{\mathrm{T}}$, 이름만!). + - **펜로즈 4조건** ($A^{+}$ 의 신분증, 식 (5)): + - 📌 원문 식 (5) (p.273): [🔗 PDF 위치](((6a856047-b418-487d-873c-2967c2407488))) + - ![펜로즈 조건 (p.273)](../assets/4장_p273_식5_펜로즈_조건.png) + - $$AA^{+}A = A, \quad A^{+}AA^{+} = A^{+}, \quad (A^{+}A)^{\mathrm{T}} = A^{+}A, \quad (AA^{+})^{\mathrm{T}} = AA^{+}$$ + - 읽는 법: 앞의 둘 = "서로가 서로를 되돌린다(갈 수 있는 데까지)", 뒤의 둘 = "$A^{+}A, AA^{+}$ 는 대칭 = **직교** 사영" (4.2 의 $P^{\mathrm{T}} = P$ 판정 그대로!). 이 4개를 만족하는 행렬은 **유일**하고 그것이 $A^{+}$. 정체 모를 후보를 검증하는 도구로 쓴다 (연습문제 2·6). + - (읽을거리) **결합행렬 예** (p.274~276): 3.5의 그래프 결합행렬($5 \times 4$, $r = 3$)에 $A = CR$ 공식을 적용해 $A^{+}$ 를 실제로 구해 보이는 예. "A의 행 합 = 0 → $A^{+}$ 의 열 합 = 0" 처럼 $A$ 와 $A^{+}$ 의 4공간이 좌우로 뒤바뀌는 것을 구경하는 것이 포인트. 키르히호프 전류법칙($A^{\mathrm{T}}y = 0$)과의 연결은 교양으로. +- ### ❓ 스스로 점검 + - 좌역·우역·유사역행렬은 각각 어떤 행렬이 갖는가? 각각 "무엇이 부족한 상황"의 처방인가? + - 좌역: 세로로 긴 열 독립 행렬 ($r = n$). **존재**가 부족한 상황(해 0 또는 1개)의 처방: 최소제곱. + - 우역: 가로로 긴 행 독립 행렬 ($r = m$). **유일성**이 부족한 상황(해 ∞개)의 처방: 최단 해. + - 유사역: 모든 행렬. 둘 다 부족하면 두 처방 동시 적용 (최소노름 최소제곱해). + - 좌역행렬의 공식은? $A^{+}b$ 가 최소제곱해인 이유는? + - $A^{+} = (A^{\mathrm{T}}A)^{-1}A^{\mathrm{T}}$ (열 독립이라 $A^{\mathrm{T}}A$ 가역). $A^{+}A = I$, 단 $AA^{+}$ 는 열공간 사영행렬 $P$. + - $A^{+}b = (A^{\mathrm{T}}A)^{-1}A^{\mathrm{T}}b$ 는 정규방정식 $A^{\mathrm{T}}A\hat{x} = A^{\mathrm{T}}b$ 의 해 그 자체다. + - 우역행렬의 공식은? $A^{+}b$ 가 최단 해인 이유는? + - $A^{+} = A^{\mathrm{T}}(AA^{\mathrm{T}})^{-1}$ (행 독립이라 $AA^{\mathrm{T}}$ 가역). $AA^{+} = I$, 단 $A^{+}A$ 는 행공간 사영행렬. + - $A^{+}b$ 는 무한한 해 중 행공간에 있는 유일한 해 = 영공간 성분 0. $\|x\|^2 = \|x^{+}\|^2 + \|x_n\|^2$ (피타고라스)라 영공간 군살이 없는 해가 최단. + - $A^{\mathrm{T}}A$ 와 $AA^{\mathrm{T}}$ 의 크기와 가역 조건은? + - 둘 다 **항상 정사각**: $A^{\mathrm{T}}A$ 는 $n \times n$, $AA^{\mathrm{T}}$ 는 $m \times m$ (내적 표). + - $A^{\mathrm{T}}A$ 가역 ⟺ 열 독립 ($r = n$) / $AA^{\mathrm{T}}$ 가역 ⟺ 행 독립 ($r = m$). + - $A^{+}$ 의 정의를 행동강령 3줄로 말하라. 그 정의가 가능한 근거는? + - $y = b + z$ (열공간 + 좌영공간)로 분해했을 때: $A^{+}b = x$ ($b = Ax$ 인 행공간의 유일한 $x$), $A^{+}z = 0$, 따라서 $A^{+}y = x$. + - 근거: "$A$ 는 행공간 → 열공간의 가역변환" (4.1). $A^{+}$ 는 그 역 + 복원 불가능한 좌영공간은 0. + - $A^{+}A$ 와 $AA^{+}$ 의 정체는? 진짜 역행렬과의 차이는? + - $A^{+}A$ = 행공간 사영행렬 ($n \times n$), $AA^{+}$ = 열공간 사영행렬 ($m \times m$). + - 역행렬은 $I$ 를 만들지만 유사역은 "갈 수 있는 최선"인 사영까지 간다. $r = n$ 이면 $A^{+}A = I$, $r = m$ 이면 $AA^{+} = I$ 로 퇴화. +- ### ✏️ 연습문제 풀이 (솔루션 참고) + - **문제 1**: 세 행렬의 유사역행렬을 구하라: $A_1 = \begin{bmatrix} 1&1 \\ 1&1 \end{bmatrix}$, $A_2 = \begin{bmatrix} 2&0 \\ 0&4 \\ 0&0 \end{bmatrix}$, $A_3 = \begin{bmatrix} 2&0&0 \\ 0&4&0 \end{bmatrix}$. + id:: c7d24191-44a3-4d31-a17a-450ef991aa54 + - 🔁 개념 상기: 세 행렬이 정확히 [2단계] 표의 세 유형이다. $A_1$ 은 랭크 부족(둘 다 부족), $A_2$ 는 세로(좌역), $A_3$ 는 가로(우역). 유형 판정 → 맞는 공식 선택이 풀이의 전부. + - $A_1$ (정사각이지만 $r = 1 < 2$, 한쪽 역도 없음): 랭크 1이라 $A_1 = cr^{\mathrm{T}}$ 꼴 ($c = r = (1,1)$). + - 문제 2의 랭크 1 공식으로: $A_1^{+} = \frac{rc^{\mathrm{T}}}{(r^{\mathrm{T}}r)(c^{\mathrm{T}}c)} = \frac{1}{2 \cdot 2}\begin{bmatrix} 1&1 \\ 1&1 \end{bmatrix} = \frac{1}{4}\begin{bmatrix} 1&1 \\ 1&1 \end{bmatrix}$. + - 감각 확인: $A_1(1,1) = (2, 2)$, 즉 $(1,1)$ 방향에서 $A_1$ 은 2배로 불린다. 되돌리기는 그 방향에서 $\frac{1}{2}$ 배여야 하는데, 실제로 $A_1^{+}(2,2) = \frac{1}{4}(4,4) = (1,1)$ ✓ (행렬 계수 $\frac{1}{4}$ 에 $A_1$ 자신의 2배가 합성되어 배율 $\frac{1}{2}$). + - $A_2$ (세로, $r = n = 2$ → 좌역): 대각이라 지름길이 보인다. **역수 취하고** ($\frac{1}{0}$ 은 0), **모양은 전치**: + - $$A_2^{+} = \begin{bmatrix} 1/2&0&0 \\ 0&1/4&0 \end{bmatrix} \qquad \text{검산: } A_2^{+}A_2 = I_2 \ ✓,\ A_2A_2^{+} \neq I_3$$ + - 공식으로 해도 같다: $A^{\mathrm{T}}A = \mathrm{diag}(4, 16)$ → $(A^{\mathrm{T}}A)^{-1}A^{\mathrm{T}}$ = 위와 동일. + - $A_3$ (가로, $r = m = 2$ → 우역): $A_2$ 의 전치니까 답도 전치의 관계: + - $$A_3^{+} = \begin{bmatrix} 1/2&0 \\ 0&1/4 \\ 0&0 \end{bmatrix} \qquad \text{검산: } A_3A_3^{+} = I_2 \ ✓$$ + - 핵심 문장: 유사역 계산의 1단계는 공식이 아니라 **유형 판정** (세로/가로/랭크부족). 대각 꼴이면 "역수 + 전치 모양 + $\frac{1}{0}$ 은 0"으로 즉답. + - **문제 2**: 랭크 1인 $A = cr^{\mathrm{T}}$ 의 유사역행렬이 $A^{+} = \frac{rc^{\mathrm{T}}}{(r^{\mathrm{T}}r)(c^{\mathrm{T}}c)}$ 임을 펜로즈 조건으로 확인하라. + - 🔁 개념 상기: 랭크 1 행렬(3장의 $uv^{\mathrm{T}}$)의 되돌리기. 검증 도구는 [8단계] 펜로즈 조건. 그리고 $\frac{rr^{\mathrm{T}}}{r^{\mathrm{T}}r}$ 꼴이 4.2의 직선 사영행렬이라는 것. + - 1단계, 두 곱부터 계산 (여기서 이미 그림이 보인다): + - $$A^{+}A = \frac{rc^{\mathrm{T}} \cdot cr^{\mathrm{T}}}{(r^{\mathrm{T}}r)(c^{\mathrm{T}}c)} = \frac{r(c^{\mathrm{T}}c)r^{\mathrm{T}}}{(r^{\mathrm{T}}r)(c^{\mathrm{T}}c)} = \frac{rr^{\mathrm{T}}}{r^{\mathrm{T}}r}$$ + - 이것은 $r$ 방향(= 행공간!) 사영행렬. 마찬가지로 $AA^{+} = \frac{cc^{\mathrm{T}}}{c^{\mathrm{T}}c}$ = $c$ 방향(= 열공간) 사영행렬. [7단계]의 $P_{\text{row}}, P_{\text{col}}$ 이 랭크 1 버전으로 눈앞에 나타났다. 대칭 ✓ (조건 3·4 통과). + - 2단계, 조건 1: $AA^{+}A = (cr^{\mathrm{T}})\frac{rr^{\mathrm{T}}}{r^{\mathrm{T}}r} = c\frac{(r^{\mathrm{T}}r)r^{\mathrm{T}}}{r^{\mathrm{T}}r} = cr^{\mathrm{T}} = A$ ✓ (가운데 스칼라 $r^{\mathrm{T}}r$ 약분, 결합법칙 트릭). + - 3단계, 조건 2: 같은 수법으로 $A^{+}AA^{+} = A^{+}$ ✓. + - 핵심 문장: 랭크 1의 유사역 = "전치 모양($rc^{\mathrm{T}}$)에 두 크기 제곱으로 나누기". 그리고 $A^{+}A, AA^{+}$ 가 행·열공간 사영행렬이 되는 것을 손으로 목격하는 문제. + - **문제 4**: $A = \begin{bmatrix} 1&0&0 \\ 0&1&0 \end{bmatrix}$ 의 행공간·열공간은? $A^{\mathrm{T}}$ 와 동일한 4공간을 갖는 $A^{+}$ 는? + - 🔁 개념 상기: [6단계] "$A^{+}$ 와 $A^{\mathrm{T}}$ 는 4공간이 같다" + 4.4 요약표의 $Q^{+} = Q^{\mathrm{T}}$. + - 공간 파악: 행공간 = $\mathbb{R}^3$ 안의 $xy$-평면 (행 $(1,0,0), (0,1,0)$ 이 생성), 열공간 = $\mathbb{R}^2$ 전체. + - $A^{+}$ 계산: 가로($r = m = 2$)라 우역인데, **행들이 정규직교**라서 $AA^{\mathrm{T}} = I_2$ → $A^{+} = A^{\mathrm{T}}(AA^{\mathrm{T}})^{-1} = A^{\mathrm{T}} \cdot I = A^{\mathrm{T}}$. + - $$A^{+} = A^{\mathrm{T}} = \begin{bmatrix} 1&0 \\ 0&1 \\ 0&0 \end{bmatrix}$$ + - 감각 확인: 이 $A$ 는 "3차원 벡터의 $z$ 성분 버리기"다. 되돌리기는 "$z$ 자리는 모르니 0으로 채워 복귀" = 전치가 하는 일 그대로 (4.2 문제 14의 그 좌표 부분공간!). + - 핵심 문장: 행(또는 열)이 정규직교면 $A^{+} = A^{\mathrm{T}}$. 4.4 의 $Q^{+} = Q^{\mathrm{T}}$ 가 그 특수 케이스였다. + - **문제 6**: $P$ 가 사영행렬일 때 ($P^{\mathrm{T}} = P = P^2$), $P^{+} = P$ 임을 펜로즈 조건으로 보여라. + - 🔁 개념 상기: [8단계] 펜로즈 4조건에 "후보 = $P$ 자신"을 넣어보는 검증 연습. + - 조건 1·2 ($A^{+}$ 자리에 $P$): $PPP = P^3 = P^2 \cdot P = P \cdot P = P^2 = P$ ✓ (두 식 모두 같은 계산). + - 조건 3·4: $(PP)^{\mathrm{T}} = (P^2)^{\mathrm{T}} = P^{\mathrm{T}} = P = P^2 = PP$ ✓. + - 왜 직관적으로 당연한가: 사영행렬은 자기 열공간 안에서는 **아무것도 안 하는** 행렬이다 (이미 도착한 벡터는 그대로). "되돌릴 것"이 없으니 되돌리기도 자기 자신. 죽인 성분(여공간)은 유사역도 똑같이 0으로 보내는 게 규칙이고. + - 핵심 문장: 사영행렬의 유사역은 자기 자신 ($P^{+} = P$). "사영은 자기 세계 안에서 항등"이라는 문장의 유사역 버전. + - **문제 11**: 왜 $A^{+}$ 의 유사역행렬은 항상 $A$ 인가? ($(A^{+})^{+} = A$) + - 🔁 개념 상기: [6단계] 그림 4.11. $A^{+}$ 의 역할을 다시 뒤집으면 무엇이 되는지 생각하는 개념 마무리 문제. + - 길 ① (4공간 관점): $A^{+}$ 는 열공간 → 행공간의 가역변환(+ 좌영공간은 0). 그것의 유사역은 "행공간 → 열공간의 가역변환 + 영공간은 0" 이어야 하는데, 그 일을 하는 행렬이 정확히 $A$ 자신이다. + - 길 ② (펜로즈 관점): 4조건 $AA^{+}A = A$, $A^{+}AA^{+} = A^{+}$, 대칭 2개는 $A$ 와 $A^{+}$ 를 **맞바꿔도 똑같은 문장**이다. 즉 "$A^{+}$ 의 유사역" 자격 요건을 $A$ 가 그대로 통과하고, 펜로즈 조건의 해는 유일하므로 $(A^{+})^{+} = A$. + - 핵심 문장: 되돌리기의 되돌리기 = 원본. 정의(4공간 역할)로도, 신분증(펜로즈 조건의 대칭성)으로도 같은 결론. diff --git a/Major Study/pages/4장 직교성.md b/Major Study/pages/4장 직교성.md index 2b365f1..6d9a209 100755 --- a/Major Study/pages/4장 직교성.md +++ b/Major Study/pages/4장 직교성.md @@ -17,4 +17,31 @@ deck:: Logseq/전공공부/선형대수학 - [[4.1 벡터와 부분공간의 직교성]] - [[4.2 직선과 부분공간 위로의 사영]] - [[4.3 최소제곱근사]] - - [[4.4 정규직교기저와 그람-슈미트 과정]] \ No newline at end of file + - [[4.4 정규직교기저와 그람-슈미트 과정]] + - [[4.5 행렬의 유사역행렬]] +- ## 4장에 대한 고찰 + - 책의 고찰(p.277~278)은 "직교벡터와 직교행렬이 이 책에서 가장 중요하다"는 선언과 함께 사실 9개를 나열한다. 4장에서 증명까지 끝낸 것(❶~❹)과 뒤 장의 예고(❺~❾)로 갈라 읽으면 깔끔하다. + - **4장에서 증명까지 끝낸 사실들**: + - ❶ $\|Qx\| = \|x\|$ (크기 보존), ❷ $(Qx)^{\mathrm{T}}(Qy) = x^{\mathrm{T}}y$ (내적·사잇각 보존): 4.4 [3단계]의 한 줄 증명. + - ❸ $Q$ 의 열공간 위로의 사영행렬은 $P = QQ^{\mathrm{T}}$: 4.4 [4단계] (역행렬이 증발한 사영). + - ❹ $Qx = b$ 의 최소제곱해는 $\hat{x} = Q^{\mathrm{T}}b = Q^{+}b$: 4.4 와 4.5 의 접점 ($Q^{+} = Q^{\mathrm{T}}$). + - **뒤 장의 예고편** (수위 조절: 지금은 이름만 알면 된다): + - ❺ 직교행렬의 거듭제곱과 곱도 직교행렬이다 (행렬식 얘기는 5장에서). + - ❻ 대칭행렬 $S$ 의 고유벡터는 정규직교할 수 있다 (6장). 최소제곱의 심장이던 $S = A^{\mathrm{T}}A$ 가 그 주인공이 된다. + - ❼ 모든 행렬의 특이벡터는 정규직교한다 (7장 SVD): 행공간의 정규직교기저 $v$ 들이 열공간의 직교벡터 $Av$ 들로 가는, 4장 큰 그림의 완성형. + - ❽ 모든 가역행렬 = (직교행렬 $Q$)(대칭행렬 $S$), ❾ 모든 가역행렬 = (직교행렬 $Q$)(상삼각행렬 $R$): ❾가 4.4 그람-슈미트의 $QR$ 이고 ❽는 6장 이후의 선물. + - 💡 **4장 전체를 꿰는 한 줄**: "직교는 분해가 되는 직각이다". 직교여공간이라 $b = p + e$ 로 남김없이 쪼갤 수 있었고(4.1~4.2), 그 분해가 해 없는 문제의 최선(4.3 최소제곱)과 공짜 좌표(4.4 정규직교기저)와 만능 되돌리기(4.5 유사역행렬)를 차례로 만들었다. + - 절별 한 줄 회고: + - 4.1 네 부분공간은 두 쌍의 직교여공간이다 (기본정리 2, $x = x_r + x_n$). + - 4.2 수선의 발이 최선이다: 오차 ⊥ 부분공간, $P = A(A^{\mathrm{T}}A)^{-1}A^{\mathrm{T}}$. + - 4.3 해가 없으면 $b$ 를 $p$ 로 바꿔 풀어라: $A^{\mathrm{T}}A\hat{x} = A^{\mathrm{T}}b$ (기하·대수·미분적분이 같은 답). + - 4.4 직교 기저면 좌표가 내적으로 공짜다: $Q^{\mathrm{T}}Q = I$, $A = QR$. + - 4.5 역행렬이 없어도 되돌리기는 있다: $A^{+}$ = 최소노름 최소제곱 기계. + - 💡 이번 장에서 스스로 뚫은 질문들 (복습 때 이 목록만 훑어도 4장이 되살아난다): $e = b - p$ 방향의 관례, 사영은 1배·반사는 2배, $x$ 와 $\hat{x}$ 의 신분 차이, $Q^{\mathrm{T}}Q$ 대 $QQ^{\mathrm{T}}$, $R$ 아래 삼각형의 0이 계산 결과인 이유, $q$ 좌표계와 원래 열 좌표계 갈아타기, 좌역 = 최소제곱·우역 = 최단 해. +- ## 이 단원 핵심문제 + - 이 장을 다시 공부할 때 아래 문제들을 먼저 풀어보면 이해가 살아있는지 확인할 수 있다 (클릭하면 해설로 이동). + - [4.1 문제 3](((d997718e-f907-4e85-a9fd-dbac6a734e05))) + - [4.2 문제 20·21](((bb759e27-0133-4e4c-92f9-71b596ea233f))) + - [4.3 문제 1](((4415f73f-00e3-45df-bc5a-7a09308ee15b))) + - [4.4 문제 15](((a7769ee9-c57b-4564-866c-c7f89b7389af))) + - [4.5 문제 1](((c7d24191-44a3-4d31-a17a-450ef991aa54))) diff --git a/Major Study/pages/hls__4장_직교성_1786280501491_0.md b/Major Study/pages/hls__4장_직교성_1786280501491_0.md index 97422d3..3105529 100755 --- a/Major Study/pages/hls__4장_직교성_1786280501491_0.md +++ b/Major Study/pages/hls__4장_직교성_1786280501491_0.md @@ -643,4 +643,123 @@ file-path:: ../assets/4장_직교성_1786280501491_0.pdf hl-color:: yellow id:: 6a82ea1f-56a2-4be1-abac-4cb47a06c55d hl-type:: area - hl-stamp:: 1786964511365 \ No newline at end of file + hl-stamp:: 1786964511365 +- 모든 행렬 A는 유사역행렬 A⁺를 갖는다. 유사역행렬은 역행렬, 좌역행렬, 우역행렬을 모두 포함한다. + ls-type:: annotation + hl-page:: 63 + hl-color:: yellow + id:: 6a855f6d-02af-464b-ad44-6cdb61f7973c +- [:span] + ls-type:: annotation + hl-page:: 63 + hl-color:: yellow + id:: 6a855f7a-997f-4ca8-95fb-d572477bd8e8 + hl-type:: area + hl-stamp:: 1787125625873 +- 행렬 A의 랭크 r이 r < m과 r < n을 만족하면 무엇을 기대할 수 있는가 + ls-type:: annotation + hl-page:: 63 + hl-color:: yellow + id:: 6a855fcf-ec62-4b22-a9ec-e7d5d880455e +- [:span] + ls-type:: annotation + hl-page:: 63 + hl-color:: yellow + id:: 6a855fda-fea7-4d81-972e-04718e48d53c + hl-type:: area + hl-stamp:: 1787125722381 +- x⁺가 영공간 성분으로 0을 갖는다 + ls-type:: annotation + hl-page:: 63 + hl-color:: yellow + id:: 6a855fe0-2eb2-4a14-91bf-9a54a70ea013 +- [:span] + ls-type:: annotation + hl-page:: 64 + hl-color:: yellow + id:: 6a855ff4-89f2-4e1d-a036-be210e4b63ab + hl-type:: area + hl-stamp:: 1787125748462 +- [:span] + ls-type:: annotation + hl-page:: 64 + hl-color:: yellow + id:: 6a856002-40c1-44eb-acc5-594e6e5f8c2a + hl-type:: area + hl-stamp:: 1787125762067 +- [:span] + ls-type:: annotation + hl-page:: 65 + hl-color:: yellow + id:: 6a85600c-3cac-4ba4-9f76-bc8e55a4ff1c + hl-type:: area + hl-stamp:: 1787125772423 +- 행공간의 두 벡터 x와 X가 열공간의 동일한 벡터 Ax = AX로 갈 수는 없다는 것 + ls-type:: annotation + hl-page:: 66 + hl-color:: yellow + id:: 6a85601e-b15a-432c-a104-43aa309f4d34 +- 열공간의 모든 벡터 b = Ax는 행공간의 단 하나의 벡터 x로부터 나온다. 그러면 A⁺b = x이다. + ls-type:: annotation + hl-page:: 66 + hl-color:: yellow + id:: 6a85602c-bc22-4eb5-86c2-1079716484c5 +- [:span] + ls-type:: annotation + hl-page:: 67 + hl-color:: yellow + id:: 6a856034-75b4-4025-8463-6bbce9302041 + hl-type:: area + hl-stamp:: 1787125811831 +- [:span] + ls-type:: annotation + hl-page:: 67 + hl-color:: yellow + id:: 6a85603b-9d37-4456-b37c-0732cb7b80d3 + hl-type:: area + hl-stamp:: 1787125818753 +- [:span] + ls-type:: annotation + hl-page:: 67 + hl-color:: yellow + id:: 6a856040-be47-4a73-a52b-d486b76c8d2f + hl-type:: area + hl-stamp:: 1787125824274 +- [:span] + ls-type:: annotation + hl-page:: 67 + hl-color:: yellow + id:: 6a856047-b418-487d-873c-2967c2407488 + hl-type:: area + hl-stamp:: 1787125831688 +- 모든 행렬 A는 행공간에서 열공간으로(즉 r차원에서 r차원으로)의 가역변환이다 + ls-type:: annotation + hl-page:: 69 + hl-color:: yellow + id:: 6a856065-6171-48c3-88a1-51dd03f8ea10 +- [:span] + ls-type:: annotation + hl-page:: 69 + hl-color:: yellow + id:: 6a85606a-40c4-4d49-be45-8b3997d0b687 + hl-type:: area + hl-stamp:: 1787125866574 +- [:span] + ls-type:: annotation + hl-page:: 69 + hl-color:: yellow + id:: 6a856077-68be-4fbd-b9af-716de18b6602 + hl-type:: area + hl-stamp:: 1787125878829 +- A⁺와 Aᵀ는 동일한 행공간과 열공간을 갖는다. 둘의 차이점은 A⁺A가 행공간 위로 사영하는 대칭인 사영행렬이라는 것이다 + ls-type:: annotation + hl-page:: 69 + hl-color:: yellow + id:: 6a856080-57ec-455d-8096-8c361f5939fc +- [:span] + ls-type:: annotation + hl-page:: 69 + hl-color:: yellow + id:: 6a856088-3271-41fd-b568-8a0c1b029b59 + hl-type:: area + hl-stamp:: 1787125895702 \ No newline at end of file

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