Math416_Rotations

# Math416_Rotations - Math 416 - Abstract Linear Algebra Fall...

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Math 416 - Abstract Linear Algebra Fall 2011, section E1 Orthogonal matrices and rotations 1 Planar rotations Deﬁnition: A planar rotation in R n is a linear map R : R n R n such that there is a plane P R n (through the origin) satisfying R ( P ) P and R | P = some rotation of P R ( P ) P and R | P = id P . In other words, R rotates the plane P and leaves every vector of P where it is. Example: The transformation R : R 3 R 3 with (standard) matrix 1 0 0 0 cos θ - sin θ 0 sin θ cos θ is a planar rotation in the yz -plane of R 3 . Proposition 1: A planar rotation is an orthogonal transformation. Proof: It suﬃces to check that R : R n R n preserves lengths. For any x R n , consider the unique decomposition x = p + w with p P and w P . Then we have k Rx k 2 = k Rp + Rw k 2 since R is linear = k Rp + w k 2 since R is the identity on P = k Rp k 2 + k w k 2 since Rp w = k p k 2 + k w k 2 since R | P is a rotation in P = k p + w k 2 since p w = k x k 2 . ± Proposition 2: A linear map R : R n R n is a planar rotation if and only if there is an orthonormal basis { v 1 ,...,v n } of R n in which the matrix of R is cos θ - sin θ 0 sin θ cos θ 0 I . (1) 1

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Proof: ( ) If there is such an orthonormal basis, then consider the plane P := Span { v 1 ,v 2 } . We have R ( P ) P because the lower-left block is 0, and R | P is a rotation of P , because of the top-left block. Moreover R satisﬁes Rv i = v i for i 3 so that R is the identity on Span { v 3 ,...v n } = P . The last equality holds because the basis { v 1 ,...,v n } is orthogonal. ( ) Assume
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## This note was uploaded on 12/16/2011 for the course MATH 416 taught by Professor Frankland during the Fall '11 term at University of Illinois at Urbana–Champaign.

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Math416_Rotations - Math 416 - Abstract Linear Algebra Fall...

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