Identifying transformations in Homework 5 involves analyzing changes in a figure’s position, orientation, and size. Key transformations include translations (sliding), reflections (flipping), rotations (turning), and dilations (resizing). Comparing vertex coordinates before and after the move helps determine the specific transformation applied.
Math 21b: Linear Algebra Spring 2018
Homework 5: Transformations in geometry
This homework is due on Wednesday, February 7, respectively on Thursday February 8, 2018.
1 a) Find the reflection matrix at the line y + x = 0 in the plane.
b) Find the 2 × 2 rotation dilation matrix which rotates by 45
◦
counter clockwise and scales by a factor
√
8.
c) Find the rotation dilation matrix which rotates around the
origin by 60
◦
clockwise and scales by a factor 14. d) Find the
projection matrix onto the line x − y = 0 in the plane.
Solution:
a) Look at the image of the basis vectors
0 −1
−1 0
.
b)
2 −2
2 2
.
c) (14/2)
1
√
3
−
√
3 1
.
d) Look at the projection of the basis vectors to get A =
1
2
1
2
1
2
1
2
.
2 Name the following transformations and give a reason by stating
a feature. Choose from Dilation, Shear, Rotation, Projec-
tion, Reflection, Reflection Dilation or Rotation Di-
lation.
a) A =
1 −2
0 1
, b) A =
0.4 0
0 0.4
, c) A =
0 0
0 1
, d) A =
3 −4
4 3
, e) A =
4 3
3 −4
, f) A =
0 1
1 0
, g) A =
0 1
−1 0
,
1
Solution:
a) A horizontal shear
b) A dilation by a factor 0.4.
c) The transformation is a projection onto the y-axis,
d) The transformation is a rotation dilation.
e) The transformation is a reflection dilation.
f) The transformation is a reflection in the line x = y,
g) The transformation is a rotation by 90 degrees clockwise
3 Match the matrices A, B, C, D, E with the transformations a)-e):
a) rotation around a line, b) orthogonal projection onto a line, c)
reflection about a line, d) reflection about a plane, e) orthog-
onal projection onto a plane. A =
1
9
1 −2 2
−2 4 −4
2 −4 4
,
B =
1
5
−5 0 0
0 3 4
0 4 −3
, C =
1
9
5 −4 −2
−4 5 −2
−2 −2 8
, D =
1
5
3 0 4
0 5 0
−4 0 3
.
E =
1
3
2 −2 −1
−2 −1 −2
−1 −2 2
,
Solution:
a) D,
b) A,
c) B,
d) E,
e) C. (You can see that after row reduction)
2
4 Assume a
2
+b
2
= 1. One of the three transformations is a rotation,
the other is a reflection about a line, the third is an orthogonal
projection onto a line. Which is which? Find the inverse in the
case of rotation and reflection.
a) C =
a b
b −a
. b) A =
a −b
b a
. c) B =
a
2
ab
ab b
2
. d) For a
2 × 2 matrix A =
a b
c d
, the determinant is defined as det A =
ad − bc. Find the determinant of a shear, of a rotation, of a
reflection about a line, of reflection in the origin, a projection onto
the x axis, a rotation-dilation matrix with parameters (a, b).
Solution:
a) Reflection,
b) Rotation,
c) Orthogonal projection,
d) Shears and rotations have determinant 1. Reflections about a line
have determinant −1. Plane reflections about a point have deter-
minant 1 since they are equivalent to 180
◦
rotations. The rotation-
dilation matrix with parameters (a, b) has determinant a
2
+ b
2
.
5 The matrix multiplication introduced in the next lecture gives
the entry ij of AB as the dot product of the i’th row of A with
j’th column of B. Verify that the product of a reflection dilation
a b
b −a
with an other reflection dilation
c d
d −c
is a rotation
dilation. What is the rotation angle and what is the scaling factor?
Discuss this with somebody else to make sure you understand this
geometrically.
3
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