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Let T be the linear transformation from tex2html_wrap_inline47 to tex2html_wrap_inline47 given by multiplication by a tex2html_wrap_inline51 matrix A. We will see in Chapter 6 that the linear transformation is a rotation of tex2html_wrap_inline47 about some line through the origin (the axis of rotation), if and only if A satisfies the following two conditions:

(i) tex2html_wrap_inline59 (in other words, the transpose of A is equal to the inverse of A) and

(ii) tex2html_wrap_inline65 .

  1. Show that the following two are matrices of rotations (check conditions (i) and (ii) above):

    tex2html_wrap_inline67

  2. Consider a rotation given by some matrix A. Explain why any vector in the axis line of the rotation must be an eigenvector of A with eigenvalue 1.
  3. Find a non-zero vector spanning the axis line of each of the two rotations given by the matrices A and B above. Note: For the axis line of B, you will probably need to use the identity tex2html_wrap_inline84 .




Eyal Markman
Thu Nov 29 09:00:03 EST 2001