Adv Alegbra HW Solutions 59

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.

Ask a homework question - tutors are online