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) $\P..
이 포스트에서 $V, W$는 모두 $F$-벡터공간으로 취급한다. 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 $AB = BA = I_n$. The matrix $B$ is called the inverse of $A$ and is denoted by $A^{-1}$. Isomorphism Definition 2. We say that $V$ and $W$ are isomorphic, denoted $V \cong W$, if $\exists T \in \mathcal{L}(V, W)$ such that $T$ ..
Left-multiplication transformation Definition 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 1 Theorem 1. Let $A, B \in M_{m \times n}(F)$. Then we have the following properties: (a) $L_A \in \mathcal{L}(F^n, F^m)$ (b) $[L_A]_..
Kronecker delta Definition 1. We define the Kronecker delta $\delta_{ij}$ by $\delta_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j. \end{cases}$ Identity matrix Definition 2. The $n \times n$ identity matrix $I_n$ is defined by $(I_n)_{ij} = \delta_{ij}$. Remark Remark. Let $A \in M_{n \times n}(F).$ Then $A$ is a diagonal matrix $\Longleftrightarrow A_{ij} = \delta_{ij} A..
이 포스트에서 $V, W, Z$는 모두 유한차원 $F$-벡터공간으로 취급한다. 함수의 합성은 보통 $g \circ f$로 표기하는데, linear transformation의 경우 $gf$로 표기하도록 하자. Theorem 1 Theorem 1. Let $T, U_1, U_2 \in \mathcal{L}(V, W)$, and let $U \in \mathcal{L}(W, Z)$. Then (a) $UT \in \mathcal{L}(V, Z)$. (b) If $UT$ is injective, then so is $T$. (c) If $UT$ is surjective, then so is $U$. (d) IF $T$ and $U$ are bijective, then so is $UT$. Introductio..
