College Algebra Exam Review 3

College Algebra Exam Review 3 - 13 1.4 SYMMETRIES AND...

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Unformatted text preview: 13 1.4. SYMMETRIES AND MATRICES We can arrange 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. Then certain axes of symmetry will coincide with the coordinate axes. For example, we can orient the rectangle in the plane so that the axis of rotation for r1 coincides with the x –axis, the axis of rotation for r2 coincides with the y –axis, and the axis of rotation for r3 coincides with the z –axis. The rotation r1 leaves the x –coordinate of a point in space unchanged and changes the sign of the y – and z –coordinates. We want to compute the matrix that implements the rotation r1 , so let us recall how the standard matrix of a linear transformation is determined. Consider the standard basis of R3 : 23 23 23 1 0 0 O O O e1 D 405 e2 D 415 e3 D 405 : 0 0 1 If T is any linear transformation of R3 , then the 3-by-3 matrix MT with O O O columns T .e1 /; T .e2 /, and T .e3 / satisfies MT x D T .x / for all x 2 R3 . Now we have O O O r1 .e1 / D e1 ; r1 .e2 / D O O e2 ; and r1 .e3 / D O e3 ; so the matrix R1 implementing the rotation r1 is 2 3 1 0 0 1 05 : R1 D 40 0 0 1 Similarly, we can trace through what the rotations r2 and r3 do in terms of coordinates. The result is that the matrices 2 3 2 3 10 0 1 00 05 and R3 D 4 0 1 05 R2 D 4 0 1 00 1 0 01 implement the rotations r2 and r3 . Of course, the identity matrix 2 3 100 E D 40 1 0 5 001 implements the nonmotion. Now you can check that the square of any of the Ri ’s is E and the product of any two of the Ri ’s is the third. Thus the matrices R1 ; R2 ; R3 , and E have the same multiplication table (using matrix multiplication) as do the symmetries r1 ; r2 ; r3 , and e of the rectangle, as expected. Let us similarly work out the matrices for the symmetries of the square: Choose the orientation of the square in space so that the axes of symmetry for the rotations a, b , and r coincide with the x –, y –, and z –axes, respectively. ...
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