Chapter 3
Linear equations
3.1. The matrix exponential
We begin with the study of the autonomous linear rst-order system
x(t) = Ax(t),
x(0) = x0 ,
(3.1)
where A is an n by n matrix. Here, as usual, we write Ax for the matrix
product whose components are g
Rectangular Matrices
Spring 2011
I am very grateful to my student Yara Abdallah who typed these lecture
notes.
Contents
1 Multiplication
2
2 Adjoint
2
3 The eigenvalue decomposition of an arbitrary m n matrix
3.1 Construction and Proof . . . . . . . . . .
Chapter 6 Notes
The pertinent question regarding civil rights is not whether the government
has the authority to treat different people differently; it is whether such
difference in treatment are reasonable
Suspect classification: classifying people on th
Math 5101
HOMEWORK 9
1. Recall the complex roots of 1.
a) Show that the n complex solutions of the equation z n = 1, called the nth
roots of unity, are zk = exp[2ik/n], k = 0, 1, . . . , n 1.
k
b) z1 is called the primitive nth root of unity. Show that zk
Math 5101
Homework 12
Name(s):
1. If every entry in an orthogonal matrix is either
the matrix?
23
69
2. Write the rank one matrix M =
1
4
or 1 , how big is
4
as xyT and write M T in the
same form.
3. Consider M =
111
,b=
111
0
. Find the pseudoinverse M +
Chapter 5 Notes
School authorities can use dogs to detect drugs in schools and that these
officials can conduct a reasonable search of you if they have reasonable
suspicion
Each state would decide that for itself, in its own constitution
Civil liberties a
MAT 510 Homework Assignment
Homework Assignment 4
Due in Week 4 and worth 30 points
Discuss one (1) project where you used a problem-solving approach to address what turned out to be
common-cause variation, or where you used a process improvement approach
MAT 510 Homework Assignment
Homework Assignment 3
Due in Week 3 and worth 30 points
The following data consists of the actual time used and potential (the best time possible for this review
process) to complete each step in the review process. The actual
MAT 510 Homework Assignment
Homework Assignment 7
Due in Week 8 and worth 30 points
The experiment data in below table was to evaluate the effects of three variables on invoice errors for a
company. Invoice errors had been a major contributor to lengtheni
MAT 510 Homework Assignment
Homework Assignment 5
Due in Week 6 and worth 30 points
The data in below table lists country code and the order to remittance (OTR) time for hardware / software
installations for the last 76 installations (from first to last).
MAT 510 Homework Assignment
Homework Assignment 8
Due in Week 9 and worth 30 points
Suppose the number of equipment sales and service contracts that a store sold during the last six (6)
months for treadmills and exercise bikes was as follows:
Treadmill
Ex
Math 5101
Homework 11
1. Consider the quadratic form on R3
F (x) = x2 2x2 + x2 + 4x1 x2 + 8x1 x3 + 4x2 x3
1
2
3
a) Find an orthonormal basis u1 , u2 , u3 , and the numbers 1 , 2 , 3 so that
3
if x =
3
yi ui
2
i yi
then F (x) =
i=1
i=1
b) Find an orthogona
5.1 Introduction 251
ination steps produced the exact answer in a nite time. (Or equivalently, Cramers
rule gave an exact formula for the solution.) In the case of eigenvalues, no such
steps and no such formula can exist, or Galois would turn in his grav
September 12, 2012
EIGENVALUES AND EIGENVECTORS
RODICA D. COSTIN
Contents
1. An example: linear dierential equations
2. Eigenvalues and eigenvectors: denition and calculation
2.1. Denitions
2.2. The characteristic equation
2.3. Geometric interpretation of
5101 (AU 2012)
Homework 1
Name(s):
Updated Homework policy: if you work in groups for your homework assignments
(up to three students in a group) please submit only one write-up, signed by all members
of the group. All co-signers will receive the same gra
Math 5101
Due Friday Sept 14
HOMEWORK 2
1. Show that the set of 22 matrices with real entries, with usual addition
and multiplication with scalars, is a real vector space. Find its dimension
and a basis.
2. For each n denote by Pn the vector space of poly
Math 5101
Due Friday Sept 21
HOMEWORK 3
Solve, write up solutions and hand in for grading the
problems 2.6.1, 2.6.2, 2.6.5, 2.6.7, 2.6.13, 2.6.16 posted on
Carmen,
in the module II. Problems from Strangs book, 3rd ed.,
topic Problems Ch2.
Solve more probl
Math 5101
Whenever you use symbolic calculation software attach a print out and
explain what you calculated and why. For Problem 4. also include the signicance of the columns of the transition matrix obtained.
HOMEWORK 4
1. For each of the following matri
Math 5101
Due Friday Oct 5
HOMEWORK 5
Solve, write up solutions and hand in for grading
all problems from 5.1.1 to 5.1.6,
also 5.1.7, 5.1.11, 5.1.13, 5.1.14, 5.1.15, 5.1.17, 5.1.18. posted on
Carmen,
in the module II. Problems from Strangs book, 3rd ed.,
Math 5101
HOMEWORK 7
1. Recall the complex roots of 1:
a) Show that the n complex solutions of the equation z n = 1 are
zk = exp[2ik/n], k = 0, 1, . . . , n 1 (called the nth roots of unity).
k
b) Show that zk = z1 . (z1 is called the primitive nth root o
Math 5101
Homework 11
due Friday Nov. 30, 2012
1. Show that the pseudoinverse of a vector x Cn is
x+ =
0,
1
x
if x = 0
x , if x = 0
2
2. Find the SVD and the pseudoinverse of the following matrices:
M=
1
1
2 2
L=
10 2 10 2
5 11 5 11
3. Let u1 , . . . , ur