Archimedean Property Archimedean Property. Let $a, b \in \mathbb{N}$. Then $\exists n \in \mathbb{N}$ such that $na > b$. 해석학에서 언급하는 아르키메데스의 성질의 정수론 버전이며, $a, b$를 자연수로 가져온 것 말고 차이는 없다. Proof. Suppose that $\forall n \in \mathbb{N}, na \leq b$. Let $S := \{b - na \, | \, n \in \mathbb{N}\}$. Since $b - na > 0$, $\emptyset \neq S \subseteq \mathbb{N} \cup \{0\}$. Then by Well-Ordering Principle, $..
Well-Ordering Principle Well-Ordering Principle. Let $\emptyset \neq S \subseteq \mathbb{N} \cup \{0\}$. Then $\exists a \in S$ such that $a \leq b, \forall b \in S$. 정렬 원리라고도 하며, 집합 내에서 최소 원소의 존재성을 보장해준다.
Equivalence of the system of linear equations Definition 1. Two systems of linear equations are called equivalent if they have the same solution set. Theroem 1 Theorem 1. Let $Ax = b$ be a system of $m$ linear equations in $n$ unknowns, and let $C$ be an invertible $m \times m$ matrix. Then the system $(CA)x = Cb$ is equivalent to $Ax = b$. Proof. Denote $K$ and $K_C$ the solution set to $Ax = b..
Reduced Row Echelon Form Definition 1. A matrix is said to be in reduced row echelon form(RREF) if the following three conditions are satisfied: i) Any row containing a nonzero entry precedes any row in which all the entries are zero. ii) The first nonzero entry in each row is the only nonzero entry in its column. iii) The first nonzero entry in each row is 1 and it occurs in a column to the rig..
Homogeneous System Definition 1. A system $Ax = b$ is said to be homogeneous if $b = 0$. Otherwise it is said to be nonhomogeneous. 번역하면 homogeneous는 '동차', 즉 차수가 같다는 말이다. $b$는 방정식에서 상수항에 해당되고, 그 외의 항들은 모두 차수가 1인 미지수들이 곱해져 있다. 즉 $b$에 해당하는 항들을 제외하면 모두 미지수의 차수가 같으므로, 만일 상수항이 0이라면 상수항에 동일한 차수의 미지수를 곱한 것으로 생각할 수 있으므로 시스템 자체를 차수가 같은, 즉 동차 연립방정식이라고 볼 수 있다. Theorem 1 Theorem 1. Let $Ax = \mathbf{0}$ be ..
A System of Linear Equations Definition 1. The system of equations $$a_{11}x_1 + \cdots + a_{1n}x_n = b_1 \\ \vdots \\ a_{m1}x_1 + \cdots + a_{mn}x_n = b_m,$$ where $a_{ij}, b_i \in F$ and $x_j$ are $n$ variables taking value in $F$, is called a system of $m$ linear equations in $n$ unknowns over $F$. 번역하면 선형 연립방정식이며, $A = \begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \text{} & \vdots \\ ..
Inverse of a Matrix Definition 1. Let $A \in M_{n \times n}(F)$. Then $A$ is invertible if $\exists B \in M_{n \times n}(F)$ such that $AB = BA = I$. The matrix $B$ is called the inverse of $A$, denoted $A^{-1}$. 위와 같이 정의되는 역행렬을 구하는 방법은 여러가지가 있으나, 여기서는 elementary operation을 이용하여 구하는 방법만을 다룬다. Augmented Matrix Definition 2. Let $A \in M_{m \times n}(F)$ and $B \in M_{n \times p}(F)$. Then the aug..
행렬의 랭크는 주어진 행렬을 다루기 쉬운 꼴로 변환함으로써 쉽게 계산해 낼 수 있다. 행렬을 RREF로 변환하는 데 성공했다면 자연스럽게 linearly independent column들을 쉽게 찾아낼 수 있으므로, 단순히 그 개수를 셈으로써 행렬의 랭크를 계산할 수 있다. Theorem 1. The rank of any matrix equals the maximum number of its linearly independent columns. Proof. Let $A \in M_{m \times n}(F)$. Consider $B := \{L_A(e_1), ..., L_A(e_n)\} = \{[A]^1, ..., [A]^n\}$ where $[A]^i$ is the $i$th column of $A$..
Theorem 1 Theorem 1. Let $T \in \mathcal{L}(V, W)$ and $U \in \mathcal{L}(W, Z)$ where $V, W$ and $Z$ are finite-dimensional vector space, and let $A, B$ be matrices such that $AB$ is defined. Then (a) rank$(UT) \leq$ rank($U$), (b) rank$(UT) \leq$ rank($T$), (c) rank$(AB) \leq$ rank($A$), (d) rank$(AB) \leq$ rank($B$). Proof. Let $A = [U]_{\beta}^{\gamma}, B = [T]_{\alpha}^{\beta}$, where $\alp..
Rank of Matrix Definition 1. Let $A \in M_{m \times n}(F)$. We define the rank of $A$, denoted rank($A$), to be the rank of $L_A: F^n \longrightarrow F^m$. Theorem 1 Theorem 1. Let $T \in \mathcal{L}(V, W)$, and let $\beta$ and $\gamma$ be oredered bases for $V, W$, respectively. Then rank($T$) = rank($[T]_{\beta}^{\gamma}$). Proof. Define $A = [T]_{\beta}^{\gamma}$. Then rank($T$) = rank($L_A$)..