Collaboration Policy: CP-1
Due: Class period on 17 Sep.
Name:
Homework 2: Fun with Linear Transforms
In this homework you will practice some basic Linear Algebra calculations to prepare you for the
upcoming exam. Each student should solve these problems solo, with no help from anybody else. You
should do all work by hand -- no calculating devices of any kind other than a pencil or pen. You may use
the course “RST Transform” handouts as reference.
Assume each set of numbers below represents a single vector. Find the length of the vector and
indicate whether it is orthonormal. Show your work. Reduce square roots and retain the exact value of
your length calculations.( e.g. √3 , 2√2 )
1. (1, 0, 0, 0)
Length
______
Orthonormal? yes/no
√3
√3
√3
1
3. ( ,
, −1)
2
2
Length
______
Orthonormal? yes/no
1
5. ( , , )
2
2 2
Length
______
Orthonormal? yes/no
2. (3, 4, 5)
Length
______
Orthonormal? yes/no
√2
−√2
1
4. ( ,
, )
2
2
2
Length
______
Orthonormal? yes/no
6. (1, -1, 1, 0)
Length
______
Orthonormal? yes/no
Find the determinants of the following matrices. Show all your work. (Hint – writing neatly will increase
both your speed and accuracy.)
10
8. �
1
𝑎
7. �
𝑐
𝑏
�
𝑑
𝑎
9. �𝑑
𝑔
𝑏
1
ℎ
𝑐
0�
𝑖
1
0
11. �
0
0
0
1
0
0
0 2
0 4
�
1 1
0 1
−1
�
20
√3
. 5 0⎤
⎡ 2
⎥
10. ⎢
√3
⎢−.5 2 0⎥
⎣ 0
0 1⎦
1 .5 2
1 3 1
12. �
.5 1 2
0 0 0
2
3
�
4
1
Collaboration Policy: CP-1
Due: Class period on 17 Sep.
Name:
2
𝑥′
�𝑦′� = �0
0
1
13. This problem uses the matrix on the right to transform the
shape shown below in the diagram. Do all of the following:
1. Find the equations for x’=f(x,y) and y’=g(x,y)
2. Calculate the four transformed vertices (x’, y’)
3. Plot the transformed shape on the diagram
x’ = ______________
(x,y)
(5, 5)
(5, 10)
(10, 10)
(10, 5)
(x’, y’)
Y
y’ = _______________
X
Create matrices that can perform the requested transforms:
14. Rotate 45 degrees counter-clockwise.
𝑥′
16. Scale by 2 along the x-axis, -1 along the
y-axis, and 1/5 along the z-axis.
20. Multiple the two given transform
matrices together to form a single
transform matrix.
𝑥′
�𝑦′� = �
𝑧′
𝑥
� �𝑦 �
𝑧
𝑥′
�𝑦′� = �
1
17. Translate -10 units along the y-axis
19. Translate -5 units along the z-axis, then
scale by 5 along all axes, then translate
+5 units along the z-axis.
�𝑦′� = �
1
𝑥
� �𝑦 �
1
𝑥
� �𝑦 �
1
𝑥′
�𝑦′� = �
1
15. Rotate 150 degrees clockwise.
18. Translate +12 units along the x-axis,
then rotate 90 degrees counterclockwise.
0 0 𝑥
. 5 0� �𝑦�
0 1 1
𝑥′
�𝑦′� = �
1
𝑥′
𝑦′
� ′� = �
𝑧
1
1 0
�0 1
0 0
��
10 2 0
5 � �0 3
1 0 0
𝑥
� �𝑦 �
1
��
��
0
0� = �
1
𝑥
� �𝑦 �
1
𝑥
𝑦
�� �
𝑧
1
�