Mathematics/Linear Algebra

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 ..

Similar Definition 1. Let $A, B \in M_{n \times n}(F)$. We say that $B$ is similar to $A$ if $\exists Q \in M_{n \times n}$ such that $Q$ is invertible and $B = Q^{-1}AQ$. Property Property. Let $A, B \in M_{n \times n}(F)$ be the similar matrices. Then (a) $A$ and $B$ have the same characteristic polynomial. Proof. (a) Since $A$ and $B$ are similar, $\exists$ invertible $Q \in M_{n \times n}(F)..

이 포스트에서 $V$는 유한차원 $F$-벡터공간으로 취급한다.    $V$의 기저 $\beta = \{x, y\}, \beta' = \{x' ,y'\}$이 주어졌을 때 임의의 벡터 $v \in V$는 각각의 기저를 사용해 좌표 벡터 $[v]_{\beta}, [v]_{\beta'}$으로 표현 가능하다. 이때 좌표를 변환하는, 즉 두 좌표 벡터 사이의 관계식을 구할 수 있다.Introduction    $[x']_{\beta} = \begin{pmatrix} a \\ b \end{pmatrix}, [y']_{\beta} = \begin{pmatrix} c \\ d \end{pmatrix}$라고 가정하자. 즉 $$x' = ax + by \\ y' = cx + dy \\ \Longrightarrow \begin..

이 포스트에서 $V, W$는 모두 $F$-벡터공간으로 취급한다.Theorem 1 (The Fundamental Theorem of Linear Algebra)Theorem 1. Let dim($V$) = $n$ and dim($W$) = $m$, and let $\beta, \gamma$ be ordered bases for $V, W$, respectively. Then the function $\Phi: \mathcal{L}(V, W) \rightarrow M_{m \times n}(F)$, defined by $\Phi(T) = [T]_{\beta}^{\gamma}$ for $\forall T \in \mathcal{L}(V, W)$, is an isomorphism.Proof. (1) $\Phi$..

이 포스트에서 $V, W$는 모두 $F$-벡터공간으로 취급한다.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 $AB = BA = I_n$. The matrix $B$ is called the inverse of $A$ and is denoted by $A^{-1}$.IsomorphismDefinition 2. We say that $V$ and $W$ are isomorphic, denoted $V \cong W$, if $\exists T \in \mathcal{L}(V, W)$ such that $T$ is i..

Left-multiplication transformationDefinition 1. Let $A \in M_{m \times n}(F)$. We denote by $L_A$ the mapping $L_A: F^n \longrightarrow F^m$ defined by $\mathsf{L}_A(x) = Ax, \forall x \in F^n$. We call $\mathsf{L}_A$ a left-multiplication transformation.Theorem 1Theorem 1. Let $A, B \in M_{m \times n}(F)$. Then we have the following properties: (a) Every left-multiplication is linear.(b) $L_A \..