Math 319
Problem Set 1: Isometries of R2
Lie Groups
This problem set is mostly a linear algebra and geometry review, but it is also
related to your first reading. Please read “The Parable of the Surveyors,” pages 1-5
of Spacetime Physics by Taylor and Wheeler.
For problems (1) – (9) that follow, assume that
• T : R2 → R2 is a linear transformation.
• T (0, 0) = (0, 0). (In fact, every linear transformation takes the zero vector to
the zero vector. Can you prove this?)
• For any two points P and Q in R2 , the distance from T (P ) to T (Q) is the same
as the distance from P to Q.
Such a linear transformation is called an isometry of R2 .
1. Suppose the distance from P to (0, 0) is d. Explain why T (P ) must lie on a
circle of radius d centered at the origin. (Don’t do this algebraically; just reason
geometrically.)
2. Use the result of (1) to explain why T (1, 0) and T (0, 1) must each lie on the circle
of radius 1 centered at the origin.
3. Assume T (1, 0) = (cos(α), sin(α)), as indicated in the figure below.
Use the assumptions about T and the result of (2) to explain why there are just
two possible locations for T (0, 1). Mark these two points on the figure above, and
determine the coordinates of each of these points in terms of the angle α. (A listing
of trigonometric identities is provided at the end of this problem set. Use them as
needed to simplify the coordinates.)
4. Each choice for T (0, 1) in (3) determines a matrix for T . Write the two matrices,
explaining which is which. You should get the two matrices:
·
¸
·
¸
cos(α) − sin(α)
cos(α) sin(α)
and
sin(α) cos(α)
sin(α) − cos(α)
1
5. For one of the two matrices in (4), T corresponds to a rotation of the plane about
the origin. Which one? (Look at the picture in (3).) What is the angle of the
rotation? What is the determinant of this matrix?
6. For the other matrix in (4), T corresponds to a reflection of the plane about a line
through the origin, sometimes called the mirror line of the reflection. What angle
does this mirror line make with the positive x axis? What is the determinant of this
matrix?
Remark: It can be shown that the results of problems 3–6 remain valid no matter
which quadrant T (1, 0) is in. (Can you show this?) Consequently, you may assume
that the two matrices in (4) always correctly describe the two possibilities for T .
7. Assume that T and S are both rotations of the plane about the origin, the matrix
of T given in terms of the angle α as in (5), and the matrix of S given similarly, but
in terms of an angle β. Use matrix multiplication to find the matrix corresponding
to the composition T ◦ S. Simplify the matrix entries using trigonometric identities,
and use the simplified matrix to show that T ◦ S is also a rotation. What is the angle
of this rotation? Does your answer make geometric sense? What can you say about
the composition in the other order, S ◦ T ?
8. Now consider the composition T ◦ S of a rotation T (expressed in terms of an angle
α as in (5)) and a reflection S (expressed in terms of an angle β, as in (6)). Describe
the result as a matrix and geometrically. Does the order of the composition matter?
Illustrate with pictures for the case when the angle of rotation is π/2 and the mirror
line for the reflection is y = x.
9. Finally, look at the composition of two reflections. What is the result? Does the
order matter? Illustrate with pictures for the case when the mirror lines are y = x
and the y-axis.
10*. This problem shows that it was unnecessary to assume that T was a linear
transformation in the beginning of this problem set. Assume only that T : R2 → R2 ,
T (0, 0) = (0, 0), and T preserves distance. Show that T is a linear transformation.
[Hint: Interpret vectors as directed line-segments of the form OP , where O = (0, 0),
and use geometric descriptions of vector addition — “parallelogram rule” — and
scalar multiplication. Argue geometrically using the pictures.]
TRIGONOMETRIC IDENTITIES
cos(0)
sin(0)
cos(−α)
sin(α + β)
cos(α + β)
sin(α − β)
cos(α − β)
=
=
=
=
=
=
=
sin(π/2) = 1
cos(π/2) = 0
cos(α)
sin(−α) = − sin(α)
sin(α) cos(β) + cos(α) sin(β)
cos(α) cos(β) − sin(α) sin(β)
sin(α) cos(β) − cos(α) sin(β)
cos(α) cos(β) + sin(α) sin(β)
2