Home

Source of most content: Gunhee Cho Eigenvector/value Def. Let $V$ be a vector space over $F$ with $dim V =n < \infty$. $v \in V \setminus \{0\}$ is called eigenvector of linear operator $T: V\rightarrow V$ if $T(v)=\lambda v$ for some eigenvalue $\lambda \in F$. Def. Linear operator $T: V\rightarrow V$ if $T(v)=\lambda v$ is diagonalizable...

Source of most content: Gunhee Cho Determinant Let $A = (a_{ij}) \in Mat_{n\times n}(F),\ B = (b_{jk}) \in Mat_{n\times n}(F)$ and $C=A \times B = (c_{ij}),$ where $c_{ij} = \sum_{k=1}^n a_{ik}b_{kj}$ and $F$ is a field. Hence, searching $A^{-1}$ is sames as finding $(b_{kj})$ s.t. $f_{ij} = \sum_{k=1}^n a_{ik}b_{kj}$ where $f_{ij}=\delta_{ij}...

Source of most content: Gunhee Cho Propostions \(\text{Given a group } G,\) $\emptyset \ne H, K \leqslant G$, then $HK= \{ hk \mid h \in H, k \in K \} \leqslant G$ iff $HK=KH$. If $H \trianglelefteq G$ and $K \leqslant G$, then $HK=KH$. If $H \trianglelefteq G$ and $K \trianglelefteq G$, then $HK \trianglelefteq G$. If $H \...

Source of most content: Gunhee Cho 1st isomorphism theorem \[\text{Let } \varphi:G \rightarrow H \text{ be a group homomorphism. Then, } \bar{\varphi}:G/\ker \varphi \cong \text{Im}\varphi \ (g\ker \varphi \mapsto \varphi(g).\nonumber\] Note that $\ker \varphi \trianglelefteq G \leadsto G/\ker \varphi$ is a group. In particular, $\bar{\varphi...

Source of most content: Gunhee Cho Group action Def. Let $G$ be a group, $X$ be a set. We call thefunction $\cdot : G \times X \rightarrow X \ ((g, x) \mapsto g\cdot x$ the group action of $G$ on $X$ if it satisfies: (Identitiy) $\forall x \in X, \ e \cdot x = x$. (Compatibility) $\forall g_1, g_2 \in G, \forall x \in X, \ (g_1g_2)...

Source of most content: Gunhee Cho Inverse of Lagrage’s theroem. \[\text{Let } G = < x > \text{ with } o(x)=n < \infty. \text{ Then, } \forall d \mid n, \ \exists! H \leqslant G \text{ s.t.} d = \mid H \mid = \mid < y> \mid = o(y). \nonumber\] Here, $o(x)$ denotes the smallest natural number $n \in \mathbb{N}$ s.t. $x^n =e$. If...

Source of most content: Gunhee Cho Subgroup Def. Given a group $(G, *), H \subseteq G$ is called a subgroup of $G$ if $(H, *)$ is a group and denoted by $H \leqslant G$. e.g.) $(\mathbb{Q}, +) \leqslant (\mathbb{R}, +)$, NB1; Let $H \leqslant (G, *)$. Since group is a monoid, $\exists! e_G \in (G, *), \exists! e_H \in (H, *)$ where $e$ impli...

Source of most content: Gunhee Cho Classification of VS Def. Given two vector spaces $V,W/F$, (1) we call $\Phi:V\rightarrow W$ a linear transform (L.T.) if $\Phi$ is a function from $V$ to $W$ and preserves a linearity, i.e., $\forall a,b \in F,\forall v,w, \in V, \ \Phi (av+bw) = a\Phi(v)+b \Phi(w)$. (2) If $\Phi^{-1}: W \rightarrow V$ is ...