Equivalence of the system of linear equationsDefinition 1. Two systems of linear equations are called equivalent if they have the same solution set.Theroem 1Theorem 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 $Ax = b$ an..

Reduced Row Echelon FormDefinition 1. A matrix is said to be in row echelon form (REF) if the following conditions are satisfied:(1) Any row containing a nonzero entry precedes any row in which all the entries are zero.(2) The first nonzero entry in each row is $1$, called the leading $1$, or the pivot.(3) Below each leading $1$ is a column of zeros.A matrix is said to be in reduced row echelon ..

Homogeneous SystemDefinition 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 1Theorem 1. Let $Ax = \mathbf{0}$ be ..

A System of Linear EquationsDefinition 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 & \ddots & \vdots \\a..

Inverse of a MatrixDefinition 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 MatrixDefinition 2. Let $A \in M_{m \times n}(F)$ and $B \in M_{m \times p}(F)$. 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 1Theorem 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..

Theorem 1Theorem 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 $\alpha, \b..

Rank of MatrixDefinition 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 1Theorem 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($L_A$) = r..

Elementary OperationDefinition 1. Let $A \in M_{m \times n}(F).$ The following operations on the rows (columns) of $A$ is called an elementary row (column) operation:- Type 1: Interchanging any two rows (columns) of $A$.- Type 2: Multiplying any row (column) of $A$ by a nonzero scalar. - Type 3: Adding any scalar multiple of a row (column) of $A$ to another row (column).    임의의 행렬이 주어졌을 때, 그 행렬의..

이 포스트에서 $V, W$는 유한차원 $F$-벡터공간으로 취급한다.Linear FunctionalDefinition 1. Let $T \in \mathcal{L}(V, F)$. Then we call $T$ a linear functional on $V$.Dual SpaceDefinition 2. We define the dual space of $V$ to be the vector space $\mathcal{L}(V, F)$, denoted by $V^*$.The double dual (or bidual) space $V^{**}$ is the dual space of $V^*$.    선형 변환 $T: V \rightarrow F$을 linear functional, 선형 범함수라고 부르고 이들을 ..