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($L_A$)..
Elementary Operation Definition 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 Functional Definition 1. Let $T \in \mathcal{L}(V, F)$. Then we call $T$ a linear functional on $V$. Dual Space Definition 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 이들..
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{pmat..
