Collaboration Policy: CP-1
Due: Class period on 27 Aug.
Name:
Homework 1: Linear Algebra
In this homework you will practice some basic Linear Algebra calculations to prepare you for upcoming
lectures. Each student should solve these problems solo, with no help from other students. You should
do all work by hand -- no calculating devices of any kind other than a pencil or pen.
In graphics, we use linear algebra to translate, rotate, and scale objects by moving their vertices. We
also use matrices to calculate the effects of light rays striking a surface. The examples below show how
to multiply a 2D position coordinate (x,y) by a 2x2 matrix:
𝑎
Example 1: �
𝑐
𝑎𝑥 + 𝑏𝑦
𝑏 𝑥
� �𝑦 � = �
�
𝑐𝑥 + 𝑑𝑦
𝑑
Example 2: �
1 2 5
1∗5+2∗6
17
�� � = �
�= � �
3 4 6
3∗5+4∗6
39
Using Examples 1 & 2 as templates, solve the following problems. Show all work. Simplify your answers.
1. �
11
00000000
5
4
�� � = �
�
6
7 12.2 19
2. �
2 1 𝑥
00000000
�� � = �
�
0 2 𝑦
6
3. �
1 0 𝑥
00000000
�� � = �
�
0 1 𝑦
6
4. �
. 707 −.707 10
0000000000000
�� � = �
�
. 707 . 707 10
6
5. �
10 0 𝑥
00000000
�� � = �
�
0 5 𝑦
6
6. �
𝑐𝑜𝑠φ −𝑠𝑖𝑛φ 𝑥
00000000000000000
� �𝑦 � = �
�
𝑠𝑖𝑛φ 𝑐𝑜𝑠φ
6
To multiply a 3D position vector (x,y,z) by a 3x3 matrix, the process is similar:
𝑎
Example 3: �𝑒
ℎ
𝑏
𝑓
𝑖
𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧
𝑐 𝑥
𝑔� �𝑦� = �𝑒𝑥 + 𝑓𝑦 + 𝑔𝑧�
𝑗 𝑧
ℎ𝑥 + 𝑖𝑦 + 𝑗𝑧
Using Example 3 as a template, solve the following problems. Show all work. Simplify your answers.
For #10, extrapolate what you know about 2x2 and 3x3 matrix-vector multiplication to solve a 4x4.
1 2
7. �4 5
7 8
3 10
00000000000
�
6� �11� = �
0
9 12
0
. 707 −.707 0 10
00000000000
9. �. 707 . 707 0� �10� = �
�
0
0
0
1 10
0
1 0
8. �0 1
0 0
1
0
10. �
0
3
0
1
0
4
0 𝑥
00000000000
�
0� � 𝑦 � = �
0
𝑧
1
0
0 0 𝑥
00000000000
0 0 𝑦
0
�� � = �
�
𝑧
1 0
5 1 1
0