【題組】

十、Let

, be a linear transformation on

for a rotation by an angle θ about a unit vector
u. Specifically, we let the matrix for

with respect to the standard basis S for

be

where

is the coordinate vector of u with respect to S, c = cos(θ)
and s = sin(θ). Furthermore, let A be the rotation matrix about the z-axis of Cartesian
coordinate system by an angle θ, i.e.,

where

be an ordered orthonormal basis for

with

and let

be the change
of-basis matrix for changing basis from B to S. Which of the following statements is/are
true?

(A)

, n1n2 +b1b2 + u1u2 = 0 and n1n3 + b1b3 +u1u3 =0.
(B) The coordinate vector of u with respect to basis B is

.
(C)

(D)

(E) The matrix for

with respect to basis B is A.