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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 3by3 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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 Fall '08
 EVERAGE
 Algebra, Matrices

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