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(b) Find the reflection of the vector about the line l through the origin that makes an angle of with the
positive x-axis.
Solution (a)
We could proceed as in Example (Example 6) of Section 4.3 and try to construct the standard matrix from the formula
where is the standard basis for . However, it is easier to use a different strategy: Instead of finding
directly, we shall first find the matrix , where
is the basis consisting of a unit vector along l and a unit vector perpendicular to l (Figure 8.5.5).
Once we have found Figure 8.5.5 . The computations are as follows:
so , we shall perform a change of basis to find
Thus
From the computations in Example 6 of Section 6.5, the transition matrix from to B is
(12)
It follows from Formula 10 that
Thus, from 12, the standard matrix for T is
Solution (b)
