Unformatted text preview: by O ( n ). (a) Show that A1 = A t for all A ∈ O ( n ). (b) Show that A t ∈ O ( n ) for all A ∈ O ( n ). (c) Show that AB ∈ O ( n ) for all A,B ∈ O ( n ). (3) Let L : R 3 → M (3) be the linear map deﬁned by L ( x ˆ i + y ˆ j + z ˆ k )) = z yz xyx Observe that the image of L is the space A (3) of 3 × 3, skewsymmetric matrices. Show that L ( u × v ) = [ L ( u ) ,L ( v )] where [ A,B ] denotes the Lie bracket of A and B , i.e. [ A,B ] = ABBA . 1...
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 Spring '08
 Staff
 Linear Algebra, Matrices, Vector Space, Complex Numbers, Orthogonal matrix, Linear map, Quaternion, 2dimensional vector space

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