## Random math: 15

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...

## Random math: 10

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 ...