11. Describe all the symmetries explicitly. a. How many are rotations (including the identity symmetry)? b. How many are reflections? c. How could you shade the figure so that the resulting figure only has three possible rotational symmetries (including the identity symmetry)?

11. Describe all the symmetries explicitly. a. How many are rotations (including the identity symmetry)? b. How many are reflections? c. How could you shade the figure so that the resulting figure only has three possible rotational symmetries (including the identity symmetry)?

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.6: The Matrix Of A Linear Transformation

Problem 4AEXP

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Question

This is a geometry question. Number 11.

11. Describe all the symmetries explicitly.
a. How many are rotations (including the identity symmetry)?
b. How many are reflections?
c. How could you shade the figure so that the resulting figure only has
three possible rotational symmetries (including the identity
symmetry)?
12. Decide whether each of the statements is true or false. Provide a
counterexample if the answer is false

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The operation between two symmetries can be thought of as a composition,with the result of the first symmetry being the starting point for thesecond symmetry. The composition symbol, °, will denote the operation between two symmetries. For example, the operation R180 ° MH would mean that, starting from the rightmost operation, the symmetry MH would be applied to the original figure first, with the result then being followed by R180, a counterclockwise rotation by 180 degrees. The result of the composition is equivalent to symmetry MV applied to the original figure (Figure 5), so we can express the final result as R180 ° MH = MV. Complete the following Cayley table: ° Apply First Apply Second ° 1 R180 MH MV 1 R180 MV MH MV The symmetries will be denoted by S, i.e., S = {1, R180, MH, MV}.ii. Explain why the system (S, °) is closed.iii. Provide at least three examples that can be used to conjecture that the system (S, °) is associative. iv.…
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Consider the example where G is the symmetries of the square, and H is the rotations of thesquare. Then the two cosets of H are { elt[1], elt[2], elt[3], elt[4] } = H = H xid, and{ elt [5], elt [6], elt [7], elt [8] ] = H • elt [5]. A set of coset representatives is{ elt[1], elt[5] } and the reader can verify that each element is uniquely expressible as aproduct of an element of H and a coset representative.