Elementary OperationDefinition 35. 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 어떤..

SimilarDefinition 34. Two square matrix $A, B \in M_{n \times n}(F)$ are said to be similar if there exists an invertible matrix $Q$ such that $B = Q^{-1}AQ$. Theorem 43Theorem 43. If $A$ and $B$ are similar, then (1) $\det(A) = \det(B)$,(2) tr$(A) = $ tr$(B)$.PropertyProperty. Let $A, B \in M_{n \times n}(F)$ be the similar matrices. Then(a) $A$ and $B$ have the same characteristic polynomial.P..

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 $[x']_{\beta} = \begin{bmatrix} a \\ b \end{bmatrix}, [y']_{\beta} = \begin{bmatrix} c \\ d \end{bmatrix}$라고 가정하자. 즉 $$x' = ax + by \\ y' = cx + dy \\ \Longrightarrow \begin..

Theorem 40 (The Fundamental Theorem of Linear Algebra)Theorem 40. 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) $\Phi$ is linear: By Theorem 29, it is ..

IsomorphismDefinition 32. We say that $V$ and $W$ are isomorphic, denoted $V \cong W$, if $\exists T \in \mathcal{L}(V, W)$ such that $T$ is invertible. Such $T$ is called an isomorphism from $V$ onto $W$.벡터공간 $V$와 $W$ 사이에 전단사 선형 변환 $T$가 존재한다는 것은 두 공간의 벡터 사이에 어떤 대응이 존재한다는 뜻이고, 연산이 보존된다는 말과 같다. 따라서 두 공간은 수학적으로 동일한 성질을 가지는 것으로 간주할 수 있고, 이때 두 공간은 isomorphic하다고 부른다. Theorem 35Theorem 35. Let $T \in ..

Left-multiplication transformationDefinition 31. 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 34Theorem 34. Let $A, B \in M_{m \times n}(F)$. Then we have the following properties: (a) Every left-multiplication is linear.(b) $L_..

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 32Theorem 32. 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 $UT$. Introduction ..

Ordered basisDefinition 27. An ordered basis for $V$ is a basis for $V$ endowed with a specific order. 기저에 순서를 부여한 것을 ordered basis, 순서기저라고 부른다. 즉 순서기저로 생각하면 $\{e_1, e_2, e_3\} \neq \{e_2, e_1, e_3\}$이다. Coordinate vectorDefinition 28. Let $\beta = \{v_1, ..., v_n\}$ be an ordered basis for V. We define the coordinate vector of $x$ relative to $\beta$, denoted $[x]_{\beta}$, by $$[x]_{\beta} ..

The nullity and rankDefinition 26. Let $T \in \mathcal{L}(V, W)$. If $N(T)$ and $R(T)$ are finite-dimensional, then we define (1) the nullity of $T$, denoted nullity($T$) := dim($N(T)$), (2) the rank of $T$, denoted rank($T$) := dim($R(T)$).$N(I_V) = \{\mathbf{0}\}, R(I_V) = V$, 그리고 $N(T_0) = V, R(T_0) = \{\mathbf{0}\}$ 임을 생각해 볼 때, 직관적으로 nullity가 클수록 rank는 작아지고, 역으로 nullity가 작을수록 rank는 커진다는 것을 눈..

The null space and rangeDefinition 25. Let $T \in \mathcal{L}(V, W)$.(a) The null space (or kernel) $N(T)$ of $T$ is the set $N(T) = \{ x \in V \,|\, T(x) = \mathbf{0} \}.$(b) The range (or image) $R(T)$ of $T$ is the set $R(T) = \{ T(x) \in W \,|\, x \in V \}$. Theorem 23Theorem 23. Let $T \in \mathcal{L}(V, W)$. Then $N(T) \leq V$ and $R(T) \leq W$.Proof. Clearly, $N(T), R(T) \neq \emptyset$. ..