\[ \boxed { \begin{array} {} & \text {preorders} & \\ & \text {reflexive:} \; x \leq x & \\ & \text {transitive:}\; x\leq y \;\&\; y\leq z \Rightarrow x\leq z & \\ \boxed { \begin{array} {} \text {partial order} \\ \text {skeletal:}\; x\leq y \;\&\; y\leq x \Rightarrow x=y \\ \text {example: power set } X\mathcal P \\ \end{array} } & \boxed { \begin{array} {} \text{intersection} \\ \text{discrete order: only $x=x$ allowed } \\ \end{array} } & \boxed { \begin{array} {} \text {equivalence relation} \\ \text {symmetric: } x\leq y \Rightarrow y\leq x \\ \text{example: congruence } m \equiv n \; (\text{mod } k) \\ \end{array} }\\ \end{array} } \]
Thursday, October 10, 2024
Wednesday, October 9, 2024
Groups in linear algebra and elementary geometry
Preliminary Draft
There are five groups that play an important role in linear algebra and elementary geometry.
The following table shows what they are and the relations between them.
Here $\bf k$ is an arbitrary commutative field (like $\R$ or $\bf C$).
\[ \boxed {\begin{array} {} \text {metric-preserving} && \text {not metric-preserving} \\ \hline \text {Isom}_n ({\bf k}) = \text {O}_n ({\bf k}) \ltimes \text {Trans}_n ({\bf k}) & \subset & \text {Aff}_n ({\bf k}) = \text {GL}_n ({\bf k}) \ltimes \text {Trans}_n ({\bf k}) \\ \cup && \cup \\ \text {O}_n ({\bf k}) & \subset & \text {GL}_n ({\bf k}) \end{array} } \]
The $\ltimes$ symbol denotes semi-direct product.
Reference: Andrew Baker, Matrix Groups
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