Homework 5 - Math 102
Sections 3.1 - 3.3
Due by 5pm Tuesday, August 20 to the dropbox in the basement of APM.
1
4
1. Find the projection p of b = 4 onto the line through 4 and the projection matrix P so that
1
2
p = P b. Draw a picture showing how b, p, a
Homework 2 - Math 102
Due by 5pm Thursday, August 8 to the dropbox in the basement of APM.
1. Show that the product of two upper triangular matrices is upper triangular. That is, if A and B are
upper triangular, show
(AB )ij = 0 if i > j.
Similarly, show
Homework 6 - Math 102
Sections 3.3 - 3.4
Due by 2pm Friday, August 23 to the dropbox in the basement of APM.
0
1
0
1
1. 3.3 #12 If V is the subspace spanned by
0 and 1
0
1
nd
(a) a basis for the orthogonal complement V .
(b) the projection matrix P ont
Homework 6 Solutions
0
1
0
1
1. 3.3 #12 If V is the subspace spanned by
0 and 1
0
1
nd
(a) a basis for the orthogonal complement V .
(b) the projection matrix P onto V .
0
1
(c) the vector in V closes to the vector b =
0 in V .
1
Solution:
1
0
1
0
0
Homework 7 - Math 102
Chapter 4
Due by 5pm Tuesday, August 27 to the dropbox in the basement of APM.
11
21
1. Find the determinant of A =
31
41
12
22
32
42
13
23
33
43
14
24
. Theres a smart way to do this!
34
44
2. 4.2 #26: If aij is i times j , sho
Homework 7 Solutions
11
21
1. Find the determinant of A =
31
41
12
22
32
42
13
23
33
43
14
24
. Theres a smart way to do this!
34
44
Solution: Recall that the row operation add a multiple of one row to another does not change the
determinant. We appl
Homework 8 - Math 102
Sections 5.1-5.3
Due by 2pm Friday, August 30 to the dropbox in the basement of APM.
1. The characteristic polynomial of a matrix A is a polynomial in ,
p() = cn n + cn1 n1 + + c1 + c0
such that p() = det(A I ). In this problem, we w
Using the Singular Value Decomposition for Image Compression
In this project, youll explore an application of a certain matrix decomposition, called the singular value
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Least Squares
In this project, we will investigate the problem of tting a curve to some data points using the method of
least squares. Our approach will be dierent than the one learned in class, but our results will be the same.
For now, suppose we want t
Google Page Rank
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Google Finds Your Needle i
Computer Graphics
In this project, we will see how linear algebra can be used in computer graphics. We wont do any
programming, but we will answer the types of questions that someone who was programming computer
graphics would need to be able to answer. I
Homework 5 - Math 102
Sections 3.1 - 3.3
Due by 5pm Tuesday, August 20 to the dropbox in the basement of APM.
1
4
1. Find the projection p of b = 4 onto the line through 4 and the projection matrix P so that
1
2
p = P b. Draw a picture showing how b, p, a
Homework 4 - Math 102
Sections 2.4 - 2.6
Due by 2pm Friday, August 16 to the dropbox in the basement of APM.
1. 2.5 #8 This problem concerns the following graph.
1
e1
e5
e4
4
e2
e6
3
e3
2
Write down the dimensions of the four fundamental subspaces for thi
1.
2.
t - h N : l
The following foreign exchange rates were printed in the newspaper, where
1; denotes US dollars, J denotes Jodanian diners, and M denotes German mark .
_d.l_ = 9.11.! __CLM_ = 1.03M.
d$ 1$ d$ 1$
Compute the following ratios
Syllabus for Math 102
Summer Session II, 2013
Course:
Math 102 (Applied Linear Algebra)
Credit Hours:
4
Prerequisites:
Math 20F or Math 31AH
Summary:
This course is a sequel to 20F, intended to give you a broader and more thorough understanding
of linear
Name
PID
Practice Midterm Exam - Math 102
70 minutes
The test covers sections 1.1 through 3.4 of your textbook, excluding section 1.7.
The actual midterm is on Thursday during class - please bring a blue book.
The format of the actual midterm will be t
Name
PID
Practice Final Exam - Math 102
3 hours
The test is cumulative, though more emphasis will be placed on later material.
The nal is on Friday from 11:30 to 2:30 - please bring a blue book.
I recommend doing this practice midterm in a test-like en
Homework 9 - Math 102
Sections 5.5-5.6
Due by 5pm Tuesday, September 3 to the dropbox in the basement of APM.
1. Explain why
3
0
0
4
is similar to
3
0
1
4
but
B=
0
i
E=
4
3+i
3i
6
3
0
0
3
2.
A=
1
1
0
1
D=
1
0
i
3
i
0
3
0
is not similar to
C=
2
1/
1/ 2
0
Homework 9 Solutions
1. Explain why
3
0
0
4
is similar to
3
0
1
4
3
0
but
0
3
3
0
is not similar to
1
3
.
30
31
31
is similar to
because
is diagonalizable. It has two dierent
04
04
04
eigenvalues and therefore two linearly independent eigenvectors. When t
Homework 8 - Math 102
Sections 5.1-5.3
Due by 2pm Friday, August 30 to the dropbox in the basement of APM.
1. The characteristic polynomial of a matrix A is a polynomial in ,
p() = cn n + cn1 n1 + + c1 + c0
such that p() = det(A I ). In this problem, we w
Homework 1 Solutions - Math 102
These are the denitions of the terms you were asked to dene in Homework 1. You should use
these denitions to study for Thursdays quiz.
A linear system of equations is any collection of equations that can be represented by a
Homework 10 - Math 102
Sections 6.1-6.3
Due by 3pm Friday, September 6 to the dropbox in the basement of APM.
1. 6.2 #4: Show from the eigenvalues that if A is positive denite, so is A2 and so is A1 .
2. The singular value decomposition uses the fact that
Homework 1 - Math 102
Due by 5pm Tuesday, August 6 to the dropbox in the basement of APM.
On a separate piece of paper, write the denition of each term.
You should try lling in what you remember without looking anything up. Then, to check your
work and ll
Homework 2 Solutions- Math 102
1. Show that the product of two upper triangular matrices is upper triangular. That is, if A and B are
upper triangular, show
(AB )ij = 0 if i > j.
Similarly, show that the product of two lower triangular matrices is lower t