Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

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ANALYTIC GEOMETRY

9. The same for the transformation of similitude which
doubles all the lengths, followed by a reflection in the axis of y.
10. Prove analytically, for the following transformations,
that the product of a transformation and its inverse is the
identical transformation, x' = x, y' = y.
(a) translations; (b) rotations;
(c) reflections; (d) one-dimensional strains.
11. Factor the transformation
x' = 4x, y '=2y,
(a) into two one-dimensional strains;
(b) into a one-dimensional strain and a transformation of
similitude.
12. Prove that the rotation about the origin through 180~also called the reflection in the origin —is the product of the
reflections in the axes.
13. Factor the transformation
x' = 4x, y' =-2y
into two one-dimensional strains and the reflection in the axis
of x. How else can it be factored?
Express each of the following transformations as the product of two or more simple transformations.
14. x =- 3x, y' -2y.
15. =-x + 2, y'= y - 3.
16. x' = 5 +2, y'= -3 - 1.
17. '= - _ 6, y, _ = Y + 3.
6. The General Affine Transformation. By the title is
meant the transformation
x'-= - ax + by + c,
(1) y' = a'x + b'y + c',
where A ab' - a'b = 0.