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($L_A$)..
