College Algebra Exam Review 4

# College Algebra Exam Review 4 - Exercise 1.4.4 ), and so...

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14 1. ALGEBRAIC THEMES Then the symmetries a and b are implemented by the matrices A D 2 4 1 0 0 0 ± 1 0 0 0 ± 1 3 5 B D 2 4 ± 1 0 0 0 1 0 0 0 ± 1 3 5 : The rotation r is implemented by the matrix R D 2 4 0 ± 1 0 1 0 0 0 0 1 3 5 ; and powers of r by powers of this matrix R 2 D 2 4 ± 1 0 0 0 ± 1 0 0 0 1 3 5 and R 3 D 2 4 0 1 0 ± 1 0 0 0 0 1 3 5 : The symmetries c and d are implemented by matrices C D 2 4 0 ± 1 0 ± 1 0 0 0 0 ± 1 3 5 and D D 2 4 0 1 0 1 0 0 0 0 ± 1 3 5 : Therefore, the set of matrices f E;R;R 2 ;R 3 ;A;B;C;D g necessarily has the same multiplication table (under matrix multiplication) as does the corresponding set of symmetries f e;r;r 2 ;r 3 ;a;b;c;d g . So we could have worked out the multiplication table for the symmetries of the square by computing products of the corresponding matrices. For example, we compute that CD D R 2 and can conclude that cd D r 2 . We can now return to the question of whether we have found all the symmetries of the rectangle and the square. We suppose, as before, that the ﬁgure (square or rectangle) lies in the .x;y/ –plane with sides parallel to the coordinate axes and centroid at the origin of coordinates. Any sym- metry takes vertices to vertices, and line segments to line segments (see
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Unformatted text preview: Exercise 1.4.4 ), and so takes edges to edges. Since a symmetry is an isom-etry, it must take each edge to an edge of the same length, and it must take the midpoints of edges to midpoints of edges. Let 2` and 2w denote the lengths of the edges; for the rectangle ` ¤ w , and for the square ` D w . The midpoints of the edges are at ˙ ` O e 1 and ˙ w O e 2 . A symmetry ± is de-termined by ±.` O e 1 / and ±.w O e 2 / , since the symmetry is linear and these two vectors are a basis of the plane, which contains the ﬁgure R . For the rectangle, ±.` O e 1 / must be ˙ ` O e 1 , since these are the only two midpoints of edges length 2w . Likewise, ±.w O e 2 / must be ˙ w O e 2 , since these are the only two midpoints of edges length 2` . Thus there are at most four possible symmetries of the rectangle. Since we have already found four distinct symmetries, there are exactly four....
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## This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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