Adv Alegbra HW Solutions 59

Adv Alegbra HW Solutions 59 - 59 2.77(i Prove that the...

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Unformatted text preview: 59 2.77 (i) Prove that the special orthogonal group SO ( 2 , R ) is isomorphic to the circle group S 1 . Solution. De fi ne ϕ : A = · cos α − sin α sin α cos α ¸ 7→ ( cos α, sin α). In Example 2.48(iii), the product of two special orthogonal matri- ces is computed, and this shows that the function ϕ is a homomor- phism. The inverse function sends ( cos α, sin α) to the matrix in the de fi nition of ϕ . (ii) Prove that all the rotations of the plane about the origin form a group under composition which is isomorphic to SO ( 2 , R ) . Solution. By Proposition 2.59, every isometry fi xing the origin is a linear transformation. The usual isomorphism between linear transformations and matrices (done in general in Chapter 4) exists here. If ² 1 = ( 1 , ), ² 2 = ( , 1 ) is the standard basis of R 2 , then ϕ corresponds to the 2 × 2 matrix whose fi rst column are the coor- dinates of ϕ(² 1 ) and whose second column are the coordinates of ϕ(² 2 ) , as in Example 2.48(iii)., as in Example 2....
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