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(a)
(b)
Use the fact stated in Exercise 17 and part (b) of Theorem 6.6.2 to show that a composition of rotations can always be
18. accomplished by a single rotation about some appropriate axis.
Prove the equivalence of statements (a) and (c) in Theorem 6.6.1.
19.
A linear operator on is called rigid if it does not change the lengths of vectors, and it is called
20. angle preserving if it does not change the angle between nonzero vectors.
(a) Name two different types of linear operators that are rigid.
(b) Name two different types of linear operators that are angle preserving.
(c) Are there any linear operators on that are rigid and not angle preserving? Angle preserving
and not rigid? Justify your answer.
Referring to Exercise 20, what can you say about if A is the standard matrix for a rigid linear
21. operator on ?
Find a, b, and c such that the matrix
22.
is orthogonal. Are the values of a, b, and c unique? Explain.
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