Unformatted text preview: ) and show that it is φ invariant. 6. Let SO (3) be the set of 3 × 3 orthogonal matrices with determinant +1. The tangent space of SO (3) at the identity is given by the set of skew-symmetric matrices of the form b ω = ( ω ) ∧ = ±-ω 3 ω 2 ω 3-ω 1-ω 2 ω 1 ² (we’ll show this later in the course). (a) Show that if v ∈ R 3 , b ωv = ω × v , where × is the cross product in R 3 . (b) Show that the tangent space T R SO (3) consists of matrices of the form b ωR where b ω is skew-symmetric. (c) Show that the ﬂow of a vector ﬁeld g ( R ) = b ωR is given by φ t ( R ) = exp( b ωt ) R where exp is the matrix exponential....
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- Fall '08
- Vector Space, Manifold, Euclidean space, Tangent space, Matrix exponential, Boothby