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--- mvg:lecture_03:start [2018/11/29 13:57] – [TRansformation between Lie Algebra and Lie Group] admin
+++ mvg:lecture_03:start [2018/12/23 18:45] (current) – [TRansformation between Lie Algebra and Lie Group] admin
@@ Line 72: / Line 72: @@
 ==== TRansformation between Lie Algebra and Lie Group ====
-Lie Algebra: $so(3) = \{\hat{w} | w \in \mathbb{R}^3\}$
+Lie Algebra: $so(3) = \{\hat{w} | w \in \mathbb{R}^3\}$ of skew symmetric matrices
-Lie Group: $SO(3)$
+Lie Group: $SO(3)$, smooth manifold and a group
+The Lie algebra $so(3)$ is the tangent space at the identity of the rotation group $SO(3)$
+algebra over a field K is a vector space over K with multiplication on the space V. Not commutative.
+Lie Brackets:
+$\left[\hat{w},\hat{v}\right] \identical \hat{w}\hat{v}-\hat{v}\hat{w} \in so(3)$
+Two Lie groups we will study: $so(3), se(3)$
+=== Exponential Map ===
+$\left\{ \dot{R}(t)=\hat{w}R(t)\right. $
+$R(0) = I$
+How does the rotation matrix change from one time within a little time step?
+$R(t) = e^{\hat{w}t} = \sum \ldots $ expansion. That solves the diff. eqn. above.
+$\exp: so(3) \rightarrow SO(3)\quad \hat{w} -> e^{\hat{w}}$