Page 300 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Find the standard matrix for the stated composition of linear operators on .
19.

(a) A rotation of 30° about the x-axis, followed by a rotation of 30° about the z-axis, followed by a contraction with

factor .

(b) A reflection about the -plane, followed by a reflection about the -plane, followed by an orthogonal projection
on the -plane.

(c) A rotation of 270° about the x-axis, followed by a rotation of 90° about the y-axis, followed by a rotation of 180°
about the z-axis.

Determine whether .
20.

(a) is the orthogonal projection on the x-axis, and is the orthogonal projection on the
y-axis. is the rotation through an angle .

(b) is the rotation through an angle , and is the rotation through an angle .

(c) is the orthogonal projection on the x-axis, and

Determine whether .
21.

(a) is a dilation by a factor k, and is the rotation about the z-axis
through an angle .

(b) is the rotation about the x-axis through an angle , and is the rotation about the z-axis
through an angle .

In the orthogonal projections on the x-axis, y-axis, and z-axis are defined by
22.

respectively.

(a) Show that the orthogonal projections on the coordinate axes are linear operators, and find their standard matrices.

(b) Show that if and is an orthogonal projection on one of the coordinate axes, then for every vector x in
the vectors are orthogonal vectors.