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# hw4 - by O n(a Show that A-1 = A t for all A ∈ O n(b Show...

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Math 136A, Spring 2008 Problem Set #4 (due 4/25/2008) (1) (a) Show that the complex numbers C may be viewed as a 2-dimensional vector space. (b) Let M (2) denote the space of 2 × 2 matrices, and let L : C M (3) be the map defined by L ( x + iy ) = x y - y x Verify that L is a linear map. What is the rank of L ? (c) Show that L satisfies the identity L ( z 1 z 2 ) = L ( z 1 ) L ( z 2 ) for all z 1 , z 2 C . (2) Let A be a n × n matrix and let A i denote the i -th row of A . We say that A is an orthogonal matrix if it satisfies the identities A i · A i = 1 and A i · A j = 0 for all i 6 = j . The set of all n × n orthogonal matrices is called the orthogonal group and is denoted by
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Unformatted text preview: by O ( n ). (a) Show that A-1 = 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 y-z x-y-x Observe that the image of L is the space A (3) of 3 × 3, skew-symmetric 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 ] = AB-BA . 1...
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