Differences

This shows you the differences between two versions of the page.

--- mvg:lecture_01:start [2018/11/19 09:05] – [Annotations / Comments / Remarks] admin
+++ mvg:lecture_01:start [2018/11/19 09:55] (current) – [Group] admin
@@ Line 12: / Line 12: @@
 Keywords
+===== Annotations / Comments / Remarks =====
 ==== Vector Spaces, Keywords ====
@@ Line 99: / Line 101: @@
 **Ring**
-$\mathcal{M(m,n)}$ is the set of all $m \times n$-matrices. $\mathcal{M(n)}$ is the set of all square matrices (
+$\mathcal{M(m,n)}$ is the set of all $m \times n$-matrices. $\mathcal{M(n)}$ is the set of all square matrices:
+$M(n) \in V$
+$\cdot: V \times V \rightarrow V$
+$+: V \times V \rightarrow V$
+==== Group ====
+Group requirements:
+  * $g_1\circ g_2 \in G$
+  * $e\circ g = g \circ e = g$
+  * $\exists g^{-1} \in G: g \circ g^{-1} = g \circ g^{-1} = e$
+  * $g_1 \circ g_2 \circ g_3 = (g_1 \circ g_2) \circ g_3 = g_1 \circ (g_2 \circ g_3)$
+Example: Rotations form a group!
+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) $