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

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

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.

Find a nonzero 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 .
Eyal Markman
Thu Nov 29 09:00:03 EST 2001