Let $G=\langle a\rangle$ be a cyclic group. Let $\eta\in\mathrm{End}\langle a\rangle$. Then "$\eta\in\mathrm{Aut}\langle a\rangle\Leftrightarrow \eta(a)$ is a generator of $\langle a\rangle$". Proof. If $\left |\langle a\rangle\right |=\in…

Properties of a (general) field $F$ Let $F$ be a field and $E/F$ a field extension. Commutative ring $R (\neq 0)$ is a field $\Leftrightarrow$ Ideals of $R$ are only $0\left(=(0)\right)$ or $R\left(=(1)\right)$ Proof. $(\Rightarrow):\,a\in…

Some properties of a group $G$ Let $K\lhd G$, $K\subset H$ be a subgroup of $G$. 1 Map \begin{align} \begin{array}{ccc} \{H\,|\,K\subset H:\text{subgroup of } G\} &\rightarrow &\{\bar{H}\,|\bar{H}:\text{subgroup of } G/K\}\\ H&\mapsto&H/K …

Shell script Extract from 3rd to 5th characters of the file name Add prefix "P-" and suffix "-S" Batch file (For MS DOS) Ex. : Delete first two characters. (abcde.txt→cde.txt) Ex. : Add "prefix" at the beginning of a file name. (abc.txt→pr…

Determinant of $r^2\mathbb{1}-3\boldsymbol{r}\otimes\boldsymbol{r}$ \begin{align} \boldsymbol{r}\otimes\boldsymbol{r} =\begin{pmatrix} x^2&xy&xz\\ yx&y^2&yz\\ zx&zy&z^2 \end{pmatrix} \end{align} This matrix has following properties: $\math…

It is very easy to derive rotation matrix. This method is applied to general cases. Derivation Step1. Calculate rotation of unit vectors Step2. Arrange two vectors Background: Why can we derive? Derivation Step1. Calculate rotation of unit…

Prefix Pressure Others Density of water: Water pressure: Prefix Name Symbol $10^n$ kilo k $10^3$ hecto h $10^2$ deca da $10^1$ $10^0$ deci d $10^{-1}$ centi c $10^{-2}$ Pressure \begin{align} 1 \mathrm{\,Pa}&=1 \mathrm{\,N/m^2}\\ \text{Atm…