What Happens to $\det(A)$ if we perform an elementary row operation on $A$ Theorem 1. Let $A \in M_{n \times n}(F)$ and $B = R(A)$, where $R$ is an elementary row operation. Then the followings hold: (a) If $R = R_{i \leftrightarrow j}$, then $\det(B) = -\det(A)$. (b) If $R = R_{ci}$, then $\det(B) = c \cdot \det(A)$. (c) If $R = R_{i + cj}$, then $\det(B) = \det(A)$.

이 포스트에서 $V$는 유한차원 $F$-벡터공간으로 취급한다. Determinant of a Linear Operator Definition 1. Let $T \in \mathcal{L}(V)$. We define the determinant of $T$, denoted $\det(T)$, to be $\det(T) = \det([T]_{\beta})$, where $\beta$ is an ordered basis for $V$. 선형 연산자 $T$의 행렬 표현의 행렬식으로 $T$의 행렬식을 정의할 수 있다. 이러한 정의는 $V$의 기저의 선택에 의존하지 않는다. Let $\beta, \gamma$ be ordered bases for $V$. By Theorem 2, we have $[T]_{\gamm..

Determinant Definition 1. The determinant of $A \in M_{n \times n}(F)$ is a scalar $$\text{det}(A) = \sum_{j=1}^n (-1)^{i+j}A_{ij}\text{det}(\widetilde{A_{ij}})$$ for some row $i$, where $\widetilde{A_{ij}}$ is the $(n-1) \times (n-1)$ matrix obtained from $A$ by deleting row $i$ and column $j$. If $n = 1$, then $\det(A) := A_{11}.$ Determinant, 즉 행렬식은 치환으로 정의되나 여기서는 흔히 라플라스 전개라고 알려진 방법으로 정의하여 잘..

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..