CS 132 Pop Quiz 5, 19 November 2013
Solutions
For Questions 1, 2 and 3, consider the following matrix A:
A=
1
2
3
4
Question 1: Find all the real eigenvalues of A. (Hint : Determine the characteristic equation rst. There are two distinct eigenvalues, one
Assignment 8: Quadratic Forms
Written by: Saber Mirzaei
Problem 1.
For any quadratic polynomial
in dimensional space of the form
is equivalent to some matrix form
cfw_
(a)
a.
where
[
b.
where
[
a.
where
[
b.
where
[
]
]
(b)
Problem 2.
(a)
a.
[
b.
[
]
]
(b
Computer Science 132 (Fall Term, 2013)
Geometric Algorithms
Solutions for EndofTerm Examination
17 December 2013
Problem 1. (39 points) Consider the 2 2 matrices A, P and Q:
3
1
0 1
10
P=
Q=
A = (1/2)
10
0 1
3 1
The rst 6 questions, (1) to (6), are each
CS 132, 10 December 2013
Review Questions for EndofTerm Exam
not including questions on orthogonal matrices, orthogonal transformations,
diagonalization, and related topics, covered over the last two weeks
which you should review on your own.
Question:
CS 132 Pop Quiz 6, 5 December 2013
Solutions
Question 1: For two n n matrices A and B , do we always have:
(A B )(A + B ) = A2 B 2
If YES, justify carefully in at most 23 lines. If NO, give a counterexample.
Answer: NO . Even though we always have:
(A B
CS 132 Pop Quiz 4, 5 November 2013
Solutions
Question 1: What are
5
0
A = 99
6
99
99
the eigenvalues of the matrix A:
0
0
7
Answer: 5, 6 and 7 .
Question 2 What is
0
1
0
B= 1
1
1
the characteristic polynomial of the matrix B :
1
1
0
Answer: 3 + 3 + 2 whic
Assignment 5: Eigenvalues and Eigenvectors
Written by: Saber Mirzaei
Problem 1.
(a) Solving equation 


(b) Solving equation 




and
and
Problem 2.
(
)
(a) We know that is an eigenvalue of a matrix iff
such that
. Also from
()
properties of deter
Assignment 6: More on Eigenvalues and Eigenvectors
Written by: Saber Mirzaei
Problem 1.
(a) Since
(b)
and
are eigenvectors of
then we have ( )
and ( )
hence:
Problem 2.
(a) Using
( ) in Matlab/Octave we get the eigenvalues are
a. For eigenvalue
and
:
(
)
Assignment 4: Determinants, Cramer's Rule, Vector Spaces
Written by: Saber Mirzaei
Problem 1.

(a)





Based on the properties of determinant, if two rows or two columns of a matrix are equal then the
determinant is zero.

(b)





If all ele
Assignment 3: Matrix Factorization and Decomposition
Written by: Saber Mirzaei
Problem 1.
If matrix
is invertible then
s.t.
[
]
[
. Assume
[
]
[
] hence:
]
Doing the multiplication we get:
[
]
[
]
[
]
Hence we must have:
is invertible
Problem 2.
lower tri
Assignment 1: Vectors an Linear Combinations
Due on Monday, Sept. 16, 2013
written by Saber Mirzaei
1
Assignment 1: Vectors an Linear Combinations
Problem 1
(a)
Solving the following system of equations and corresponding augmented matrix we get that the p
Assignment 2: Matrix Equations and Linear Independence
Written by: Saber Mirzaei
Problem 1.
The augmented matrix is:
[
]
[
]
Using rref function to find the Reduced Row Echelon form:
(
)
[
]
Translating back to equation for:
Hence:
[]
[
]
[
]
[
]
[
]
Prob
Assignment 6: More on Eigenvalues and Eigenvectors
Written by: Saber Mirzaei
Problem 1.
(a)
(
)
cfw_
(b)
(
)
Problem 2.
(a)
(
)
Now after finding
[
]
[
cfw_ and
we can find
]
(b) Following the same approach we get:
[
]
[
]
Problem 3.
(a)
Step1: Find the
CS 132
Review Questions
22 Oct 2013
Example 1
Let a1 = (1, 2, 3) and a2 = (5, 13, 3).
1. What does Spancfw_a1 , a2 dene in R3 : a line? a plane? the full space R3 ?
2. Is Spancfw_a1 , a2 a vector subspace of R3 or just a subset of R3 ?
3. Does Spancfw_a
CS 132 Pop Quiz 2, 1 October 2013
Solutions
The inverse of a n n square matrix A, denoted A1 if it exists, is a matrix such
that A A1 = A1 A = I where I is the n n identity matrix.
Question 1: The following is a 3 3 elementary matrix:
100
A = 0 1 0
501
A
CS 132 Geometric Algorithms
Homework 3
Due: Friday May 29th, 2015 by 11:59pm
All problems are from [LAA] 5th Edition, Section 1.4, pages 4042.
1. Problems 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.
2. Programming Assignment. For this assignment, you will implem