Differences

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--- mvg:lecture_01:start [2018/11/19 09:15] – [Group] admin
+++ mvg:lecture_01:start [2018/11/19 09:55] (current) – [Group] admin
@@ Line 115: / Line 115: @@
   * $e\circ g = g \circ e = g$
   * $\exists g^{-1} \in G: g \circ g^{-1} = g \circ g^{-1} = e$
-  * $g_1 \circ g_2 \circ g_3 = ($g_1 \circ g_2) \circ g_3 = $g_1 \circ (g_2 \circ g_3)
+  * $g_1 \circ g_2 \circ g_3 = (g_1 \circ g_2) \circ g_3 = g_1 \circ (g_2 \circ g_3)$
-E.g. rotations form a group!
+Example: Rotations form a group!
-Set of invertable square matrices form the General Linear Group GL(n). They are closed with respect to multiplication.
+Citation (DC):
+A group G has a matrix representation or can be realized as a matrix group if there exists an **injective** transformation:
+$R: G \rightarrow GL(n)$
+Why not surjective, i.e. bijective in total? $R(\phi) \rightarrow M$ and $R(n\cdot2\pi+\phi) \rightarrow M$ are mapped to the same element of $M \in SO(n) \subset GL(n)$.
+Set of invertible square matrices form the General Linear Group GL(n). They are closed with respect to multiplication.
 $\det(M) = 1$
+$R: G \rightarrow GL(n)$
+$R(e) = I, R(g\circ h) = R(g)R(h), \all g,h \in G$
+"It preserves the group structure!" (DC)
+$R$ is a group **homomorphism**.
+=== Affine Group $A(n)$ ===
+$L(x) = A(x) + b$
+$L: \mathbb{R}^{n+1}\rightarrow \mathbb{R}^{n+1}$
+$\left( \begin{array}{cc}
+A & \mathbf{x}  \\\\
+& 1 \\\\
+\end{array}\right) $
+===== Appendix: My MathJax Template for a Matrix =====
+$\left( \begin{array}{rrrr}
+& 0 & \cdots & 0 \\\\
+& \ddots & 0 & \vdots \\\\
+\vdots & 0 & \ddots & 0 \\\\
+& \cdots & 0 & 1
+\end{array}\right) $