iii A0 1 2 Prove Theorem 4 Let Tbe a glide refl ection with

iii. A=[0 1]. 2. Prove Theorem 4. Let Tbe a glide refl ection with axis Show that a line 1 \". Satisfies Telle if and only if Illan or 1\". 4. Let e be a line of E2. Let G be the set of isometries T of E2 satisfying Te = e. Describe the elements of G explicitly and give the group multiplication table. 5. i. Given two intersecting lines, prove that there is a rotation that takes one to the other. Is it unique? ii. Given two parallel lines, prove that there is a translation that takes one to the other. Is it unique? 6. Verify that the mapping described following Theorem 5, which associates to each affine transformation a 3 by 3 matrix, is an injective homomorphism of AF(2) into GL(3). Prove Theorem 7(i) by a direct computation using linear algebra. Prove Theorem 7(ii) by a direct computation using linear algebra.

Solution

5)i) When two lines intersect at a point say O, then an angle is formed between the lines

Let angle between two lines be theta.

Then rotation of one line through angle theta takes to the other line and the second line rotating to -theta takes the second line to the first line.

As theta is unique, this rotation is unique.

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ii) When there are two parallel lines, the perpendicular distance between them is always constant = d (say)

Then consider the perpendicular line to both cutting first line at (a,b) and second line at (c,d)

Then distance between (a,b) and (c,d) = d (always)

Consider the translation of (a,b) to (c,d)

This is unique obviously as there can be only one perpendicular from a point.

This transformation is the required one.