From 95d6c3ace68f13da692b372a435d0fde6ea2783c Mon Sep 17 00:00:00 2001 From: wikijs Date: Sat, 11 Feb 2023 18:02:55 +0000 Subject: [PATCH] docs: add all untracked content --- Sandbox.md | 49 +++ home.md | 13 + 군(수학).md | 62 ++++ 리스프.md | 16 + 모노이드.md | 19 + 모듈(수학).md | 33 ++ 벡터공간.md | 329 ++++++++++++++++++ 선형대수.md | 42 +++ 선형변환.md | 25 ++ 스킴.md | 13 + 스택(자료구조).md | 22 ++ 아벨군.md | 17 + 장수누리파크.md | 31 ++ .../장수누리파크_고추잠자리.jpg | Bin 0 -> 553985 bytes .../장수누리파크_메인.jpg | Bin 0 -> 362880 bytes 집합(수학).md | 31 ++ 큐(자료구조).md | 86 +++++ 행렬.md | 58 +++ 행렬식.md | 34 ++ 환(수학).md | 57 +++ 20 files changed, 937 insertions(+) create mode 100644 Sandbox.md create mode 100644 home.md create mode 100644 군(수학).md create mode 100644 리스프.md create mode 100644 모노이드.md create mode 100644 모듈(수학).md create mode 100644 벡터공간.md create mode 100644 선형대수.md create mode 100644 선형변환.md create mode 100644 스킴.md create mode 100644 스택(자료구조).md create mode 100644 아벨군.md create mode 100644 장수누리파크.md create mode 100644 장수누리파크/장수누리파크_고추잠자리.jpg create mode 100644 장수누리파크/장수누리파크_메인.jpg create mode 100644 집합(수학).md create mode 100644 큐(자료구조).md create mode 100644 행렬.md create mode 100644 행렬식.md create mode 100644 환(수학).md diff --git a/Sandbox.md b/Sandbox.md new file mode 100644 index 0000000..f526f49 --- /dev/null +++ b/Sandbox.md @@ -0,0 +1,49 @@ +--- +title: Sandbox +description: +published: true +date: 2020-09-22T08:57:45.548Z +tags: +editor: markdown +dateCreated: 2020-09-22T08:29:08.474Z +--- + +# Sandbox + +## testtesttest + +### dff + +#### 44444 + +##### 55555 + +test things + +---- + +```python +def fac(n): + if n == 1: return 1 + else: return n*fac(n-1) +``` + +`\frac{5, 4}` + +> Comments here! + +- [ ] An empty todo +- [x] finished + +```kroki +graphviz + +digraph { + A -> B; + B -> C; +} +``` + +## Katex + +$$\frac5 4 + \frac3 6 = \frac{21}{12}$$ \ No newline at end of file diff --git a/home.md b/home.md new file mode 100644 index 0000000..e80d221 --- /dev/null +++ b/home.md @@ -0,0 +1,13 @@ +--- +title: Welcome to My wiki +description: welcome message of my wiki +published: true +date: 2020-09-21T12:49:28.176Z +tags: +editor: markdown +dateCreated: 2020-09-21T12:49:25.655Z +--- + +# Welcom to my Wiki + +Public domain \ No newline at end of file diff --git a/군(수학).md b/군(수학).md new file mode 100644 index 0000000..7ae73a7 --- /dev/null +++ b/군(수학).md @@ -0,0 +1,62 @@ +--- +title: 군(수학)(Group) +description: Group +published: true +date: 2020-09-26T14:22:04.517Z +tags: 대수 +editor: markdown +dateCreated: 2020-09-22T07:04:46.800Z +--- + +# 군(Group) +군은 [집합](/집합(수학)) G과 [집합](/집합(수학)) G위에 다음과 같은 이항연산 $G \times G \rarr G$가 있는 것을 말한다: + +- 결합법칙이 성립한다. +- $\forall a \in G, ua = a = au$인 $u\in G$가 존재한다. +- $\forall a \in G, aa^{-1}=a^{-1}a=u$인 $a^{-1}\in G$가 존재한다. + +이때 $u$를 항등원(identity element)라고 부르고 $a^{-1}$을 a의 역원(inverse of a)라고 한다. +$|G|$는 $G$의 원소 수를 말하며, 군의 위수(Order)라고 부른다. + +## 군의 예 + +- 정수의 집합과 이항연산 +에 대해서, $(\Z,+)$는 군이다. +- 집합 $\{-1,1\}$과 이항연산 $\cdot$에 대해서 $(\{1,-1\},\cdot)$ 는 군이다. + +## 군의 사상(Morphism of group) + +군 $(G,\cdot)$의 사상은 군$(H,*)$에 대해 다음과 같은 조건을 만족하는 함수 $f:G\rarr H$이다. +$$f(a\cdot b)=f(a)* f(b)$$ + +## 순환군(Cyclic group) +어떤 군 $G$에 대해서, +$$\{a^n|a\in G,n \in \N\}$$ +를 순환군이라고 한다. 이때 $a$를 군의 생성원(generator)이라고 한다. + +## 부분군(Subgroup) + +[집합](/집합(수학)) $H$가 있어서, $H \subset G$이고 H가 G의 연산에 대해 군이면 H를 G의 부분군이라고 한다. +삽입사상 $i:H\rarr G$가 있다. + +## 관계로서 군의 표현과 자유군(Groups presentation as defining relation and free group) +aphabet이 있다. concat 연산이 있다.이것이 자유구.ㄴ +## 대칭군$S_n$과 교환군$A_n$ (Symmetric and Alternating groups) +$f:n\rarr n$를 원소로 가지는 군이다. (1 2 3 4 5) 라는 열이 있을 때, (2 1 3 4 5) 같이 치환하는 연산들로 이루어진것. + +$S_n/\Z_2\cong A_n$ + +## 변환군(Transformation groups) + +대칭군의 일반화. + +## 잉여(Cosets) +군의 원소 여럿을 묶기. 묶은 것으로 연산하기. +$h:G\rarr H, h \in \hom(G,H)$를 생각하자. + +## 핵과 상(Kernel and Image) +$\ker(h)=\lbrace a\in G|h(a)=0 \rbrace$ +$\text{img}(a)$ +### 정규부분군(Normal subgroup) +$g\in G,n \in N, gng^{-1}\in N$ +## 잉여군(Quotient Groups) + diff --git a/리스프.md b/리스프.md new file mode 100644 index 0000000..203773e --- /dev/null +++ b/리스프.md @@ -0,0 +1,16 @@ +--- +title: 리스프(Lisp) +description: 리스프 프로그래밍언어 +published: true +date: 2020-09-29T15:18:27.240Z +tags: +editor: markdown +dateCreated: 2020-09-29T15:18:24.712Z +--- + +# 리스프 +리스프(Lisp, LISP) 혹은 리습은 프로그래밍 언어의 계열로서, 오랜 역사와 독특하게 괄호를 사용하는 문법이다. + +LISP 라는 이름 자체는 "LISt Processing"(리스트 프로세싱)의 줄임말이다. 연결 리스트는 리스프의 주요 자료구조 중 하나로서, 리스프 코드는 그 자체로 하나의 리스트이다. + +코드와 데이터가 교환이 가능하다는 것은 리스프 그 자체에 있어 코드를 즉각 해석할 수 있는 능력을 준다. 전체 프로그램 코드는 S-표현식 이나 괄호로 묶인 리스트로 작성되며, 함수 호출의 경우 함수 이름 혹은 연산자가 첫 번째로 위치하여 피연산자가 이어 위치하게 된다. 예를 들면, 함수 f 가 a, b, c 라는 세 개의 피연산자를 가진 경우는 (f a b c)와 같이 표기한다. \ No newline at end of file diff --git a/모노이드.md b/모노이드.md new file mode 100644 index 0000000..3d99776 --- /dev/null +++ b/모노이드.md @@ -0,0 +1,19 @@ +--- +title: 모노이드 +description: 수학의 모노이드 +published: true +date: 2020-09-22T12:20:16.260Z +tags: 대수 +editor: markdown +dateCreated: 2020-09-22T12:20:13.496Z +--- + +# 모노이드(Monoid) +모노이드 $M$는 다음과 같은 이항연산을 가진 다음과 같은 집합 $M$을 말한다. +- 모든 $a,b,c\in M$에 대해 $a(bc)=(ab)c$ (결합법칙(Associative law)) +- $a1=1a=a$인 원소 $1\in M$이 존재한다. + +## 모노이드의 예 +## 모노이드의 사상(Morphism of monoid) +## 잉여 모노이드(Quotient monoid) +## 자유 모노이드(Free monoid) \ No newline at end of file diff --git a/모듈(수학).md b/모듈(수학).md new file mode 100644 index 0000000..8ca118b --- /dev/null +++ b/모듈(수학).md @@ -0,0 +1,33 @@ +--- +title: 모듈(Module) +description: 대수의 모듈 +published: true +date: 2020-11-06T10:44:44.161Z +tags: 대수 +editor: markdown +dateCreated: 2020-09-22T11:55:35.558Z +--- + +# 모듈(Module) +$R$-모듈 $A$는 [환(Ring)](/환(수학)) $R$과 [아벨군(Abelian Group)](/아벨군) $A$위에 이항연산 $R \times A \rarr A$이 있고, 다음과 같은 [공리(Axiom)](/공리(수학))를 따르는 $A$이다. +모든 $\kappa,\lambda \in R$과 $a,b \in A$에 대해서: +$$\begin{array}{rcl}\kappa(a+b)&=&\kappa a+\kappa b,\\ (\kappa+\lambda)a&=&\kappa a+\lambda a,\\ +(\kappa \lambda)a&=& \kappa(\lambda a),\\ 1a&=&a.\end{array}$$ +좀더 명확하게, 이러한 모듈 $A$를 왼쪽 모듈(left module)이라고 한다. 왜냐하면, 식 $\kappa a$에서 스칼라$\kappa$가 모듈의 원소 $a$의 왼쪽에 쓰여졌기 때문이다. + +## 선형변환(Linear transformations) +- main article : [선형변환](/선형변환) +모듈의 사상(Homomorphism)은 선형변환이다. + +## 부분모듈(Submodules) + +## 잉여모듈(Quotient modules) + +## 자유모듈(Free modules) + +## Biproducts + +## 쌍대모듈(Dual modules) + +지금까지 다룬 것은 스칼라가 왼쪽에 붙은 left module이다. right module은 스칼라가 오르쪽에 붙는다. +쌍대 모듈은 \ No newline at end of file diff --git a/벡터공간.md b/벡터공간.md new file mode 100644 index 0000000..2bbe1d2 --- /dev/null +++ b/벡터공간.md @@ -0,0 +1,329 @@ +--- +title: 벡터공간(Vector space) +description: 벡터공간 +published: true +date: 2020-10-24T14:41:55.900Z +tags: 선형대수, 대수 +editor: markdown +dateCreated: 2020-09-22T11:33:05.470Z +--- + +# 벡터공간(Vector space) +벡터공간은 체 $F$위의 [모듈(Module)](/모듈(수학))이다. + +어떤 벡터 리스트 $\textbf v=( v_1,v_2,\cdots, v_n)$가 있으면, 벡터 리스트를 의 일차결합을 생각할 수 있다. + +$$a_1 v_1+a_2 v_2+\cdots+a_n v_n = \sum a_i v_i$$ + +이러한 일차 결합을 선형조합이라고 한다. + +## 기저와 좌표(bases and corddinates) + +어떤 벡터 $w$를 벡터 리스트 $\textbf v$에 대해서 $w=\sum{v_n\xi_n}$ 로서 $\xi_n$으로 표현할 때, +그 리스트 $\textbf v$가 공간 $V$를 생성(span)한다는 것은 $\xi_i$으로 **적어도** 한가지로 표현한다는 것이고, +리스트 $\textbf v$가 공간 $V$의 기저(basis)라는 것은 $\xi_i$으로 **정확히** 한가지로만 표현된다는 것이고, +리스트 $\textbf v$가 공간 $V$에 선형독립(linearly independent)라는 것은 $\xi_i$으로 **최대** 한가지로 표현된다는 것이다. +가장 마지막의 선형 독립에 대한 조건을 바꾸어보면, +$$\begin{alignedat}{2} + \sum v_i \xi_i&=\sum v_i\eta_i &\Rightarrow \xi_i=\eta_i\\ + \sum v_i(\xi_i-\eta_i)&=0 &\Rightarrow \xi_i -\eta_i=0\\ +\sum v_i\lambda_i&=0 &\Rightarrow \lambda_i=0\\ +\end{alignedat}\\ +$$ +으로 영벡터에 대하여 유일하게 모두 계수가 0으로만 표현된어야 한다는 것을 의미한다. + +### Proposition 1 +$L_{\textbf v}:F^n\rightarrow V$, $L_{\textbf v}(\xi)=v_1\xi_1+\dots+v_n\xi_n$ +이란 함수를 정의하면, + +- 리스트 $\textbf v$ 가 공간 $V$를 생성(span)한다 $\Leftrightarrow$ $L_{\textbf v}$ 는 epimorphism이다. +- 리스트 $\textbf v$가 $V$에서 선형독립(lineary independence)이다 $\Leftrightarrow$ $L_{\textbf v}$는 monomorphism이다. +- 리스트 $\textbf v$가 $V$의 기저(basis)이다. $\Leftrightarrow$ $L_{\textbf v}$는 isomorphism이다. +#### 증명 +단순하게 $L_{\textbf v}$는 선형조합이므로 [선형변환](/선형변환)이다. 자세하게는 리스트 ${\bf \xi, \eta}$에 대해서, +$$L_{\textbf{v}}({\bf \xi + \eta})=\sum_i v_i(\xi_i+\eta_i)=\sum_iv_i\xi_i+\sum_iv_i\eta_i=L_{\textbf{v}}({\bf \xi})+L_{\bf v}(\eta)$$ +이고, 스칼라 $\kappa$에 대해, +$$L_{\textbf{v}}({\bf \xi}\kappa)=\sum_iv_i(\xi_i\kappa)=\sum_i(v_i\xi_i)\kappa=L_{\textbf{v}}({\bf \xi})\kappa$$ +이므로 선형변환이다. +$L_{\textbf v}$가 공간 $V$를 생성한다는 것은 어떤 벡터 $w$에 대해서, $L_{\textbf v}(\xi)=w$인 $\xi$가 있다는 것이므로, epimorphism이다. 비슷하게 선형독립이란 것은 monomorphism이고, 기저라는 것은 isomorhpism이다. + +이로부터 좌표(Coordinate)를 정의할 수 있다. + +### 좌표(Coordinate) +기저(basis) $\bf b$ 가 있어 $v=b_i\xi_i+\cdots b_n\xi_n$ 이면, +$\xi_n$는 기저가 $\bf b$인 $v$의 좌표(coordinates of $\bf v$ relative to a basis $b$)이다. +이때 리스트 $\textbf b: n\rightarrow V$는 벡터공간 $V$의 기저(basis)이다. + +### Theorem 2. +주장 1(Proposition 1)에서 $L_b:F^n\cong V$임으로, $F^n$는 [자유모듈](/모듈(수학)#자유모듈free-modules)(free module)이다. 그러므로 $L_b(\epsilon_i)=b(i)$라고 할 수 있다. + +기저(basis)가 $b$인 벡터공간(vector space) $V$은 $b$의 벡터들이 자유생성원(free generator)인 자유 F-모듈(free F-module)이다. + +### Proposition 3 +벡터 리스트가 선형종속(linearly dependent) 할때 생성원 하나 줄이면: +$v_k=v_i(-\xi_1\xi_k^{-1})+\cdots+v_{k-1}(-\xi_{k-1}\xi_k^{-1})$ +그에 따른 결과로 +#### Corollary +유한한 종류의 벡터공간은 기저를 가진다. +나눔환(division ring)에서도 성립한다. + +## 차원(Dimention) + +### THEOREM 4 + +같은 벡터공간 $V$에서 선형 독립 $m$ 벡터 리스트 $\bf{v}$있을 때, 벡터공간 $V$을 생성(span)하는 어떤 $n$ 벡터 리스트든지 $n\geq m$이다. 더 나아가, $\bf{v}$를 포함하는 $n$ 벡터 리스트는 $V$를 생성(span)할 수 있다. + +증명 : 수학적 귀납법 사용하자. + +$m=0$일 때, 결과는 자명하다. + +$m=m$일 때, $m$ 독립 벡터의 모든 리스트에 대해 성립한다고 가정하자. +이제 $\bf{v}$가 $m+1$ 선형독립벡터 리스트라고 하자. +귀납 가정에 의해서, +$(v_1,v_2,\cdots ,v_m,w_1,\cdots,w_{n-m})$ 는 $V$를 생성한다. +그러면 $v_{m+1}$도 생성 가능하다. + +$$\tag{3} v_{m+1} = v_1\xi_1+\cdots +v_m\xi_m+w_1\eta_1,\cdots,w_{n-m}$$ + +만약 $m=n$ 이면, $w_i$는 없고 그럼 $v_{m+1}$이 $v_1,\cdots,v_m$에 대해 표현되니 $v_1,\cdots,v_m,v_{m+1}$이 선형독립이라는 가설에 모순. 그래서 $m+1 \leq n$ 이다. + +Adjoint $v_{m+1}$. + +$(v_1,v_2,\cdots ,v_m,v_{m+1},w_1,\cdots,w_{n-m})$ + +이것도 여전히 $V$를 생성한다. + +위의 (3)에 의해 선형 종속이다. + +Proposition 3에 의해서 벡터 하나를 뺄 수 있다. + +$v_1,\cdots,v_{m+1}$이 선형독립이기에 $w$에서 뺼 수 있다. + +빠지는 벡터를 $w_j$라 하면, + +$(v_1,v_2,\cdots ,v_m,w_1,\cdots,w_{j-1},w_{j+1},\cdots,w_{n-m})$ + +여전히 $V$ 생성한다 그리고 containing the whole list $V$. + +수학적 귀납법 끝. 증명 끝. + +#### Corollary 1 Invariance of Dimension + +유한 타입의 벡터 공간 $V$의 아무 두 기저는 같은 수의 기저를 가지고 있다. + +증명 : $(b_1,b_2,\cdots ,b_m)$와 $(c_1,c_2,\cdots ,c_n)$를 두 $m$ 벡터 기저와 $n$ 벡터 기저라고 하자. 그러면, + +|선형독립|$V$를 생성||결론| +|:---|:---:|---:|---| +|$b$|$c$|implies|$m\leq n$| +|$c$|$b$|implies|$n\leq m$| + +그래서 $n=m$이다. 증명 끝. + +$V$의 기저의 벡터의 개수를 차원(dimension)이라고 부르고, $\dim V=n$이라고 표기한다. $\dim V$가 무한이라는 것의 의미는 $V$가 유한한 기저를 가지고 있지 않다는 것이다. 이 정의에 의해 $\dim F^n=n$ 이다. 유한한 타입의 벡터공간을 유한차원 벡터공간이라고 부른다. + +#### Corollary 2 + +$n$ 차원 벡터공간 $V$에서 + +- (i) 아무 $V$의 $n+1$ 벡터들의 리스트는 선형종속이다. +- (ii) $V$를 생성하는 $n-1$ 벡터들의 리스트는 없다. + +--- +### Theorem 5 + +In a 유한 차원 벡터 공간 $V$: + +- (i) 어떤 선형독립 벡터들 리스트를 잡든 그것은 기저의 일부분이다. +- (ii) $V$를 생성하는 벡터들 리스트는 기저를 포함한다. + +증명 : To prove (ii), Apply reduction process in proposition 3. + +As for(i), find the list of vectors spanning $V$, which must exist. + +Apply reduction process in theorem 4. + +### Corollary + +$n$ 차원 벡터공간 $V$에서 + +- (i) 아무 $V$의 $n$ 선형독립벡터들의 리스트는 기저이다. +- (ii) $V$를 생성하는 $n$ 벡터들의 리스트는 기저이다. + +증명: For(i), $v$가 $n$ 선형독립벡터 리스트라고 하자. + +그럼 $v$는 기저의 일부분이다. + +기저는 $n$개의 원소인데. (ii)도 비슷하다. + +### Proposition 6. 아무 codomain이 유한차원 $V'$인 monomorphism $s:V\rightarrow V'$는 linear left inverse를 가진다. 아무 codomain이 유한 차원 $V$인 epimorhpism $V\rightarrow V'$도 linear right inverse를 가진다. + +증명 : $V$가 $m$ 차원이고 $V'$이 $n$ 차원이라 하자. + +s가 monomorphism 이니까 $V$에서 선형독립이면 $V'$에서 선형독립이고 $m\leq n$ 이다. + +$V$에서 $m$ 벡터의 기저 $\bf b$를 고르자. + +그럼 $s({\bf b})$도 $V'$에서 선형독립이다. + +$V'$에서 $s(\bf b)$를 포함한 기저를 만들고 $(s(b_1),s(b_2),\cdots,s(b_m),b'_{m+1},\cdots,b'_n)$ 이라고 지칭하자. + +이것은 free $F$-module $V'$ 의 free generator list 이므로 + +선형 사상 $t:V'\rightarrow V$ : +$$\begin{cases} +t(s(b_i))=b_i &\text{for }i\in \bf{m}\\ +t(b'_j)=0 &\text{for each } i = m+1,\cdots,n +\end{cases}$$ + 를 정의할 수 있다. + +그래서 $(t\circ s)(b_i)=b_i$, + +그러므로 $t\circ s$ 는 무조건 identity $V\rightarrow V$ 이니 $t$ 는 left inverse 다. + +두번째 epimorphism 은 spanned 가지고 비슷하게 하면 된다. + +## 기저로 만들기(Construction for bases) + +### Proposition 7 + +$S$ 가 유한차원벡터공간 $V$의 부분공간(subspace)면, +각각의 $S$의 기저는 V 기저의 일부분이다. +그래서, $\dim S \le \dim V$; 더욱이, $\dim S \lt \dim V$ 이면 $S$는 $V$의 진부분공간(proper subspace)이다. + +Proof: $S$에서 선형독립 벡터리스트는 $V$에서도 선형독립이다. +$V$에서 선형독립인 벡터 리스트는 Theorem 5에 의해 기저의 일부분이다. +더나아가, $S$의 기저는 $V$의 모든 벡터들이 $S$안에 있을때, 전체공간 $V$의 기저이다. + +Combine 기저 v, w +$$v\vee w=(v_1,\cdots,v_m,w_1,\cdots,w_r)$$ + +### Theorem 8 + +만약 $t:V \rightarrow V'$가 유한차원정의역의 선형번환이면, + +$$\dim V = \dim (\text{Ker}\;t)+\dim (\text{Im}\;t)\qquad(10)$$ + +더 자세하게는 $\text{Ker}\;\bf t$의 기저 $v$가 $V$의 기저 $\bf v\vee w$의 일부분이면, $t\circ {\bf w}$는 $\text{Im}\;t$의 기저이다. + +증명 : Think proposition 7 + +$\bf v\vee w$에서 $v\in V$를 선형조합으로 나타내면, + +$$v=\sum v_i\xi_i+\sum w_j\eta_j\qquad(t(v_i)=0)$$ + +로 나타낼수 있다. + +$t(v)=\sum t(w_j)\eta_j$ 니까 $\bf w$는 $\text{Im}\;t$를 생성한다. + +반면에, $\sum (tw_j)\eta_j=0 \Leftrightarrow t(\sum w_j\eta_j)=0$ 이면 kernel. + +so, $\sum w_j\eta_j=\sum v_i\xi_i$인데 그러나 $\bf v\vee w$는 선형독립. 그래서 기저 $\bf v$는 선형독립. 선형독립이고 $\text{Im }V$ 생성하니 기저. 증명 끝. + + +$\text{rank }t=\dim(\text{Im }t), \text{nullity }t=\dim(\text{Ker t})$ + +(10) states that "rank plus nullity equals dimension of domain." +Sometimes $\text{Ker }t$ is called the "nullspace", and $\text{Im }t$ the "range" of t. + +#### Corollary 1. 두 벡터 공간 $V$와 $V'$이 각각의 유한차원 $n$과 $n'$을 가지고 있으면 선형사상 $t:V\rightarrow V'$의 $\text{rank }r$은 기껏해야 최대 $n$과 $n'$ 작은 것 이다. + +For each such map $t$, there is a basis $b$ of $V$ and a basis $b'$ of $V'$ such that + +$$tb_1=b',\cdots,tb_r=b_r',tb_{r+1}=0,\cdots,tb_n=0$$ + +#### Corollary 2. $V$와 $V'$가 같은 유한차원의 벡터공간이라 하자. 아무 epimorphism t 나 아무 monomorphism t는 isomorphism 이다. + +Proof: + +$\dim \text{Im }t=n$ 라는 것은 $\dim \text{Ker }t=0$ 이란 것이고 그러면 morphism $t$는 isomorphism 이다. +반대로, $\dim \text{Ker }t=0$ 라는 것은 $\dim \text{Im }t=n$ 이란 것이고 그러면 역시 morphism $t$ 가 isomorphism 이다. + +#### Corollary 3. $\dim (V/S)=\dim V-\dim S$ + +#### Corollary 4. + +$S$가 $V$의 부분공간이면 $V$의 direct summand고, $V$의 부분공간 $T$가 존재해서 $S+T=V$이고 $S\cap V=0$ 이다. 더더욱, $T\cong V/S$ 이다. + +--- + +### Proposition 9 + +만약 벡터공간 W는 두 유한차원 부분공간 $V_1$과 $V_2$의 직합이면, $V_1$의 기저 $b'$ 와 $V_2$의 기저 $b''$에 대해 $b'\vee b''$는 $W$의 기저이다. 그런 이유로 $W$는 유한차원이고 $\dim W = \dim V_1 + \dim V_2$ + +$V_1\oplus V_2$ 가 직합(direct sum)이라고 하자. 그러면, + +#### Corollary 1. $\dim(V_1\oplus V_2)=\dim V_1+\dim V_2$ + +#### Corollary 2. $\dim(S+T)=\dim S+\dim T - \dim(S\cap T)$ + +proof: + +$$\begin{aligned}\\ +\dim(S+T)&=\dim ((S+T)/(S\cap T)) + \dim(S\cap T)\\ +&=\dim (S/(S\cap T))+\dim (T/(S\cap T))+ \dim(S\cap T)\\ +&=\dim S+\dim T - \dim(S\cap T) +\end{aligned}$$ + +### Proposition 3 + +$\dim(V^{\bf m})=m\dim V \qquad(15)$ + +증명:m=1, (15) is immediate. +assume (15) for some m. +$V^{\bf m+1}$ is $f:{\bf m+1}\rightarrow V$ + +assigned to $(f',f(m+1))$ + +$V^{\bf m+1}\cong V^{\bf m}\oplus V$ + +#### Corollary $\dim[\text{Hom}(V,V')]=(\dim V)(\dim V')$ + +proof: $\text{Hom}(V,V')\cong(V')^n\qquad n=\dim V$. + +## 쌍대로 묶인 벡터공간(Dually paired vector space) +벡터 공간 $V$를 right $F$-Module 로 보면, 그 쌍대(dual)은 left $F$-Module로 볼 수 있는 $V^* = \text{Hom}_F(V,F)$ 이다. + +$f \in V^*$에 대해 생각해보자. +$$\begin{aligned} +f(v)&=f(b_1\xi_1+\cdots+b_n\xi_n)\\ + &=f(b_1)\xi_1+\cdots+f(b_n)\xi_n +\end{aligned}$$ +이니까 $f$는 $fb_1, \cdots,fb_n$에 의해 유일하게 결정된다. +$f\mapsto f\cdot \bf{b}$ 인 morphism을 생각하면 isomorphism 이다.($f:V\rightarrow F, f\cdot {\bf b} : F^{n} \rightarrow F$) +$$\begin{aligned} +\delta_{ij} &= 1 \text{ for } i=j\\ +&=0 \text{ for }i\neq j +\end{aligned}$$ +를 크로네커 델타 기호라 부른다. + +$fb_i=\lambda_i$라 하면, $f$는 $\lambda_i$에 대해 유일하게 결정된다. + +$x_i\cdot b_j = \delta_{ij}$ 인 $x_i$를 잡으면 + +$f = \sum\lambda_ix_i$ 라고 할수 있다. + +그럼 $x_1\cdots x_n$ 는 기저이다. + +이 결과로 당연히 $\dim V^*=n$이다. + +### Lemma 1. 0이 아닌 linear form $f \in V$에 대해서 적어도 $f(v)=1$ 인 $v$가 존재한다. + +증명 : 이것은 자명하다. $f \ne 0$ 이면, $f(u)\ne 0$ 인 어떤 벡터 $u$가 존재한다. 그러면 적당한 역원을 곱해서 $f(v)=1$ 인 벡터를 얻을 수 있다. + +### Lemma 2. 0이 아닌 유한차원벡터공간의 벡터 $u$ 에 대해서, 적어도 $f(u)=1$인 linear form이 존재한다. + +증명 : $u\ne 0$이면, u는 선형 독립이므로, $V$ 기저의 일부분이다. +f 를 그 기저의 dual 기저로 잡으면 된다. + +$f\in V^*$ 는 변수 $v$에 대해서 $v\mapsto f(v)$로 정의된다. 만약에 $v$를 고정하고 $f$를 변수라고 생각하면, $f\mapsto f(v)$는 $V^*\rightarrow F$ 인 함수가 된다. + +이 함수에게 $\omega(v):V^*\rightarrow F$라고 이름 붙여주자. 그럼 $\omega$는 $V\rightarrow V^{**}$ 로 대응되는 함수이다. + +### Corollary 3. 모든 유한차원 벡터공간 $V$는 자기 자신의 이중 dual $V^{**}$와 $(\omega v)(f)=f(v)$로 정의된 morphism $\omega$의해 동형이다. + +증명 : $\omega$가 선형인건 자명하다. Lemma 2는 $\text{Ker}(\omega) = 0$임을 보인다. ($V^{**}$에서 0은 $\omega({\bf 0})$이다.) 그리고 $\dim V=\dim V^*=\dim V^{**}$이니까, $\omega$는 동형사상이다. + +$V\cong V^*$는 기저를 정하고 동형사상을 만들었던 반면에,사상 $\omega: V \rightarrow V^{**}$는 어떤 기저에 대한 언급 없이 정의 되었다. 이러한 $\omega$를 natural map 이라고 함. + +Anihilator \ No newline at end of file diff --git a/선형대수.md b/선형대수.md new file mode 100644 index 0000000..b617e0b --- /dev/null +++ b/선형대수.md @@ -0,0 +1,42 @@ +--- +title: 선형대수학 +description: 선형대수학 +published: true +date: 2020-09-22T11:55:39.992Z +tags: 선형대수, 대수, linear algebra, mathmatics +editor: markdown +dateCreated: 2020-09-21T13:09:09.265Z +--- + +# 선형대수(linear algebra) + +선형대수는 $a_1x+\cdots+a_nx=b$ 같은 일차식과 [선형변환](/선형변환)에 대해서 다루는 수학의 한 갈래이다. +선형대수는 다음과 같은 내용을 포함한다. + +## 벡터공간(vector space) +- main article : [벡터공간](/벡터공간) + +현대적으로 선형대수학은 벡터공간 V에 대해서 다루어진다. 벡터공간은 단순히 [체](/체(수학)) $F$위의 [모듈](/모듈(수학))이다. +구체적으로는, 스칼라들의 [체](/체(수학)) $F$상에서의 집합 $V=\lbrace (a_1,a_2,\cdots,a_n): a_i \in F \rbrace$ 에 아래 두 개의 이항연산이 정의되고 이 연산들과 함께 $V$의 성분들이 2개의 기본 조건과 8가지의 연산법칙을 만족하면 $V$가 주어진 벡터의 덧셈 및 스칼라배 연산 과 함께 [벡터공간](/벡터공간)을 이룬다고 하고 $(V,+,\cdot)$를 벡터공간이라고 하며 그 안의 원소를 벡터라고 한다. 공리로부터 이 벡터들은 덧셈과 함께 [아벨군](/아벨군)을 형성하고 벡터에 스칼라를 곱하는 곱셈(스칼라 곱)은 아래식을 만족한다. + +$$\begin{aligned} +a(\textbf{v}+\textbf w)=a\textbf v+a \textbf w, 1\textbf v=\textbf v \\ +(a+b)\textbf v= a \textbf v+b \textbf v,(ab)\textbf v=a(b\textbf v) +\end{aligned}$$ + +이 식에서 $\textbf v$와 $\textbf w$는 공간 $V$에 속하는 벡터이고, $a,b$는 체 $F$ 에 속하는 스칼라이다. + +## 선형조합(linear combination) + +어떤 벡터 리스트 $(\textbf v_1,\textbf v_2,\cdots,\textbf v_n)$가 있으면, 벡터 리스트를 의 일차결합을 생각할 수 있다. + +$$a_1\textbf v_1+a_2\textbf v_2+\cdots+a_n\textbf v_n = \sum a_i\textbf v_i$$ + +이러한 일차 결합을 선형조합이라고 한다. +주어진 n개의 백터의 모든 일차결합에 의해 만들어지는 집합은 $V$의 부분공간(subspace)을 이룬다. 이 부분공간을 $\textbf v_1,\textbf v_2,\cdots,\textbf v_n$에 의하여 생성된(spanned) $V$의 부분공간이라고 한다. $\sum a_i\textbf v_i=0$인 일차결합에서 모든 계수가 0인 경우에만 일차결합의 값이 0이 되는 경우에, $\textbf v_1,\textbf v_2,\cdots,\textbf v_n$을 일차독립(linear independent)이라 한다. 벡터공간 $V$에 속하는 모든 벡터가 일차 독립인 $\textbf v_1,\textbf v_2,\cdots,\textbf v_n$들에 의하여 오직 한가지의 일차결합으로만 표현될 때, $\textbf v_1,\textbf v_2,\cdots,\textbf v_n$를 유한차원 벡터공간 $V$의 기저(basis)라고 한다. 기저를 이루는 벡터들은 일차독립이어야 한다. 같은 벡터공간의 두 기저벡터는 같은 수의 원소를 가지고 있는데 이 원소의 수를 그 공간의 차원, $\dim V$이라고 한다. + +## 선형변환(linear transformation or linear map) +- main article : [선형변환](/선형변환) + + + diff --git a/선형변환.md b/선형변환.md new file mode 100644 index 0000000..3447960 --- /dev/null +++ b/선형변환.md @@ -0,0 +1,25 @@ +--- +title: 선형변환 +description: linear transformation 아님 linear map +published: true +date: 2020-09-22T11:55:39.992Z +tags: 선형대수, 대수, mathmatics +editor: markdown +dateCreated: 2020-09-21T13:19:14.157Z +--- + +# 선형변환 + +$$\begin{aligned} +t(\textbf v+\textbf w)&=&a\textbf v+a\textbf w \\ +t(a\textbf v)&=&at(\textbf v) \end{aligned}$$ +선형 변환이란 다음과 같이 덧셈과 스칼라 곱셈을 보존하는 변환이다. + +이 변환은 [모듈](/모듈(수학))의 [사상](/사상(수학))이다. + +## 선형성이란 +어떤 효과가 '선형적이다' 라고 말하는 것은 그 효과가 비례의 관계가 있다는 것과, 전체 효과는 부분 효과의 합으로 나타내진다는 것을 뜻한다. +## 선형변환의 예 +$R^3$에서 회전이나 평행이동, 잡아늘리는 변환, 투영이 선형변환이다. + +미분연산자와 적분연산자는 함수 공간에서의 선형변환이다. \ No newline at end of file diff --git a/스킴.md b/스킴.md new file mode 100644 index 0000000..5f130ed --- /dev/null +++ b/스킴.md @@ -0,0 +1,13 @@ +--- +title: 스킴(Scheme) +description: +published: true +date: 2020-09-29T15:18:28.556Z +tags: +editor: markdown +dateCreated: 2020-09-29T05:16:23.743Z +--- + +# 스킴(Scheme) + +스킴은 [리스프(lisp)](/리스프)의 방언이다. \ No newline at end of file diff --git a/스택(자료구조).md b/스택(자료구조).md new file mode 100644 index 0000000..03fcac9 --- /dev/null +++ b/스택(자료구조).md @@ -0,0 +1,22 @@ +--- +title: 스택(Stack) +description: 자료구조 스택 +published: true +date: 2020-09-22T07:26:15.886Z +tags: 컴퓨터 +editor: markdown +dateCreated: 2020-09-22T07:12:51.265Z +--- + +# 스택(Stack) + +LIFO(Last in first out) + +```rust +trait Stack{ + fn push(&mut self,item :T) -> Result<()>; + fn pop(&mut self) -> Result; + fn is_empty() -> bool; + fn size() -> usize; +} +``` \ No newline at end of file diff --git a/아벨군.md b/아벨군.md new file mode 100644 index 0000000..5d65de3 --- /dev/null +++ b/아벨군.md @@ -0,0 +1,17 @@ +--- +title: 아벨군(abelian group) +description: 아벨군 +published: true +date: 2023-02-11T12:45:33.712Z +tags: 대수, mathmatics +editor: markdown +dateCreated: 2020-09-21T14:16:27.615Z +--- + +# 아벨군(abelian group) +아벨군은 교환법칙이 성립하는 [군](/군(수학))이다. 교환군(commutative group)이라고도 부른다. + +## 성질 + +아벨군$(G,+)$은 $\forall a,b \in G,a+b=b+a$이다. +어떤 아벨군은 $\Z_n$ 군의 곱으로 나타낼수 있다. \ No newline at end of file diff --git a/장수누리파크.md b/장수누리파크.md new file mode 100644 index 0000000..f985680 --- /dev/null +++ b/장수누리파크.md @@ -0,0 +1,31 @@ +--- +title: 장수누리파크 +description: 장수 누리파크 +published: true +date: 2020-10-01T09:16:25.638Z +tags: +editor: markdown +dateCreated: 2020-10-01T09:15:11.016Z +--- + +# 장수 누리파크 + +![장수누리파크_메인.jpg](/장수누리파크/장수누리파크_메인.jpg) + +4인승 자전거 탈 수 있음 + +10-01 월~금 무료. 주말 카페에서 시키면 무료로 이용 가능. + +오전 8시에 시작하고, 오후 5시에 마감한다. + +고추 잠자리원 + +![장수누리파크_고추잠자리.jpg](/장수누리파크/장수누리파크_고추잠자리.jpg) + +이 연못에 연들이 차있다. +황소 동상이 있다. + +중앙에는 꽃이 있다. + +의자가 이쁘다. + diff 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 +``` + +## 원형큐(circular queue) + +```diagram 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 +``` + +### 구현 +```rust +const QUEUE_SIZE=128; +struct CircularQueue{ + arr:[T;QUEUE_SIZE], + first: u32, + near : u32 +}; +``` +isfull +```rust + +impl CircularQueue{ + fn new() -> CircularQueue{ + CircularQueue{ + [0;QUEUE_SIZE] + } + } + fn is_full(&self) -> bool{ + (self.rear + 1) % QUEUE_SIZE == self.first; + } +} +``` + +```rust +impl Queue for CircularQueue +{ + fn is_empty(&self){ + self.near == self.first + } + fn enqueue(&mut self, item: T) -> Result<()>{ + if self.is_full(){ + Error() + } + else{ + self.rear = (self.rear +1) % QUEUE_SIZE; + self.arr[self.rear + 1] = item; + Ok(()) + } + } + fn dequeue(&mut self) -> Result{ + if self.is_empty(){ + Error() + } + else{ + self.first = (self.first + 1) % QUEUE_SIZE; + Ok(self.arr[self.rear]) + } + } +} +``` \ No newline at end of file diff --git a/행렬.md b/행렬.md new file mode 100644 index 0000000..63fec7e --- /dev/null +++ b/행렬.md @@ -0,0 +1,58 @@ +--- +title: 행렬(Matrices) +description: 행렬 +published: true +date: 2020-09-22T14:00:28.667Z +tags: 선형대수, 대수, mathmatics +editor: markdown +dateCreated: 2020-09-21T16:08:48.764Z +--- + +# 행렬(Matrices) + +행렬은 적당한 기저를 정해서 만들어진 순서쌍이다. + +## 정사각행렬(sqaure matrices) +$n \times n$ 꼴인 행렬이다. +## 대각행렬(diagonal matrix) +대각선에만 숫자가 있고, 나머지는 0인 행렬이다. +formal하게는, +$$A = \lbrack a_{ij}\rbrack (a_{ij}=0\text{ if } i\neq j)$$ +인 행렬이다. + +## 역행렬(Inverse matrices) + +행렬이 정사각행렬이면, 다음과 같은 조건을 만족하는 행렬$A$을 생각할 수 있다. +$$A^{-1}A= I = AA^{-1}$$ +이때, $A$를 invertible하다고 하고, $A^{-1}$을 A의 역행렬("$A$ invsere","the inverse of $A$")이라고 한다. + +성질들: +* 만약 $A$가 invertible하면, 선형연립방정식에 대해서 $Ax=b \iff x=A^{-1}b$. +* 행렬 A가 non-singular면 A는 invertible 하다. 반대로 A가 invertible하면 non-singular다. +* non-zero vector $\textbf x$에 대해서 $A\textbf x=0$이라면, A는 invertible하지 않다. +* 당연히 각각의 행렬 $A$에 대하여 각각의 역행렬$A^{-1}$은 유일하다. + +2x2 matrix $A =\begin{bmatrix}a&b\\ c& d\end{bmatrix}$의 역행렬 공식: +$$A^{-1}=\frac{1}{ad-bc}\begin{bmatrix} +d&-b\\-c&a\end{bmatrix}$$ + +$ad-bc$가 0이라면 역행렬은 존재하지 않는다. 여기서 $ad-bc$는 행렬 $A$의 [행렬식(Determinant)](/행렬식)이다. + +간단하게, [대각행렬](#대각행렬diagonal-matrix)의 역행렬은 +$$\begin{bmatrix} +d_1&0&\cdots &0 \\ +0 &d_2 &\cdots& 0 \\ +\vdots & \vdots&&\vdots\\ +0 &0&\cdots&d_n +\end{bmatrix}^{-1}=\begin{bmatrix} +\frac{1}{d_1}&0&\cdots &0 \\ +0 &\frac{1}{d_2} &\cdots& 0 \\ +\vdots & \vdots&&\vdots\\ +0 &0&\cdots&\frac{1}{d_n} +\end{bmatrix}$$ + +## 가우스-조던 소거법(Gauss-Jordan Elimination) + +$AX=I$인 $X$를 구하기 위해서 [기본 행 연산(Elementary operation)](/기본행연산)을 곱해서, +Upper traiangle form이나 echelon form을 만들어서 구한다. + diff --git a/행렬식.md b/행렬식.md new file mode 100644 index 0000000..4d600cc --- /dev/null +++ b/행렬식.md @@ -0,0 +1,34 @@ +--- +title: 행렬식(Determinants) +description: 행렬식 +published: true +date: 2020-09-23T13:57:53.743Z +tags: 선형대수, 대수 +editor: markdown +dateCreated: 2020-09-22T14:00:24.726Z +--- + +# 행렬식(Determinant) + +## Multilinear and alternating functions +먼저 이중선형(bilinear)에 대해서 생각해보자. +이중선형함수는 각각 변수에 대해서 선형인, 두 변수를 받는 함수이다. 예를 들어 $V$가 벡터공간이고 $V^*$가 그 듀얼이라고 하면 $e:V^*\times V \rarr F, e(f,v)=f(v)$라는 evaluation 함수에 대해 생각해보자. + +$f$를 고정하면, +$$e(f,v_1k_1+v_2k_2)=e(f,v_1)k_1+e(f,v_2)k_2$$ +왜냐하면 $f$는 선형이기 때문이다. 반대로 $v$를 고정하면, +$$e(k_1f_1+k_2f_2,v)=k_1e(f_1,v)+k_2e(f_2,v)$$ +이다. 그러므로 함수 $e$는 이중선형이다. 일반적으로 $C$와 $D$가 두 $K$-모듈이라고 하자. 그러면 +### 이중선형(Bilinear) +$C\times D$를 오른쪽 $K$-모듈 $E$로 대응시키는 $K$-이중선형함수 $h$는 다음과 같은 property를 가진 함수 $h:C\times D \rarr E$이다. +$$h(c_1k_1+c_2k_2,v)=h(c_1,v)k_1+h(c_2,v)k_2\\ +h(c,d_1k_1+d_2k_2)=h(c,d_1)k_1+h(c,d_2)k_2$$ + + +scewed + +## Determinants of matrices + +## Cofactors and cramer's rule + +## Determinants of maps \ No newline at end of file diff --git a/환(수학).md b/환(수학).md new file mode 100644 index 0000000..43a422c --- /dev/null +++ b/환(수학).md @@ -0,0 +1,57 @@ +--- +title: 환(Ring) +description: 대수에서의 환 +published: true +date: 2020-09-23T12:14:24.004Z +tags: 대수 +editor: markdown +dateCreated: 2020-09-22T12:10:21.670Z +--- + +# 환(Ring) + +환(Ring) $R = (R,+,\cdot,1)$은 다음과 같은 두 이항연산과 원소 1있는 집합 $R$이다. +- $(R,+$)는 [아벨군](/아벨군)이다. +- $(R,\cdot,1)$은 [모노이드(Monoid)](/모노이드)이다. +- 곱은 분배적이다. + +마지막 조건은 뜻하는 것은 $a,b,c \in R$인 모든 세 원소가, +$$a(b+c)=ab+ac,\text{ }(a+b)c=ac+bc$$ +를 만족한다는 것이다. + +## 환 만들기(Constructions for rings) +한 환(Ring)으로부터 여러가지 환을 만들기. +### 부분환(Subrings) + +### Function ring + +환 R과 집합 X에 대해서 $R^X$의 덧셈과 곱셈을 +$$(f+g)(x)=f(x)+g(x), (fg)(x)=f(x)g(x)$$ +라고 하면 환을 이룬다. + +### $\text{End}(A)$ + +[아벨군](/아벨군)의 endomorphism으로도 환을 만들 수 있다. +아벨군 $A$에 대해서, 그 $\text{End}(A)$를 원소로 하고 덧셈과 곱셈을 이렇게 정의하자. +$f,g\in End(A)$와 $a\in A$에 대해서, +$$(f+g)(a)=f(a)+g(a), (f\cdot g)(a)=f(g(a))$$ +라고 하면 이것이 환(Ring)이 된다. +#### 증명 + +## 잉여환(Quotient rings) + +## 정역과 체(Integral Domain and field) + +## 다항식환(polynomial rings) + +## 함수로서의 다항식(Polynomials as functions) + +## 나눗셈 알고리즘(The division algorithm) + +## 주아이디얼역(Principal ideal domains) + +## 소체(Prime fields) + +## 유클리드 알고리즘(The euclidean algorithm) + +## 교환잉여환(Commutative quotient rings) \ No newline at end of file