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
:
(
)
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
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
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, 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:
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
Assignment 2 Solution
Due 13th February using gSubmit only
1. Find a parametric equation of the line M through p and q. [Hint: M is parallel to the vector q p.
See the figure below].
Figure 1: The line through p and q
(a)
2
p=
5
3
q=
1
(b)
6
3
CS 132 at Boston University
Assignment4
Due 18th March @ 11:59pm using gSubmitonly.
1. Use LU factorization and find the inverse of the following matrix:
3 5 0 0 0
1 2 0 0 0
0 0 2 0 0
0 0 0 7 8
0 0 0 5 6
You must show all steps and first solve for L and U
CS 132
Assignment 1 Solution
Due 3rd February using gSubmit only
1. Determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system.
(a)
2
4
3
6
h
7
(b)
1
5
Solution:
2 3 h
2
(a)
4 6 7
0
1 3 2
(b)
5 h 7
3
0
1
0
3
h
CS 132 at Boston University
Assignment5
Due 2nd April @ 11:59pm using gSubmitonly.
1
1. Let P = 3
4
2
v1 = 2 v2
3
2 1
5 0 and let
6 1
8
7
= 5 v3 = 2
2
6
(a) Find a basis cfw_u1 , u2 , u3 for IR3 , such that P is the change of coordinates matrix fr
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
CS 132 at Boston University
Assignment2
Due 13th February @ 11:59pm using gSubmitonly.
1. Find a parametric equation of the line M through p and q. [Hint: M is parallel to the vector q p.
See the figure below].
Figure 1: The line through p and q
(a)
p=
2
CS 132
Assignment 3 Solution
Due 21th February using gSubmit only
1. Let T: IR3 IR3 be a transformation that projects each vector x = (x1 , x2 , x3 ) onto the plane x2 = 0
so T (x) = (x1 , 0, x3 ). Show that T is a linear transformation.
Solution: Take u
CS132
Kevin Xia
U92603848
Assignment #7
Assignment #7
1.
a. If = 0, then the following is true:
% = % = % 0 = 0
%
b. If = 0, then the following is true:
% % = % % = % 0 = 0
The following is true:
= % = % % = 0
This means that since = 0, = 0 must be t
CS132 Assignment 4
Kevin Xia
[email protected]
April 4, 2017
Assignment 4
1. First, let C be the basis v, where v = cfw_v1, v2, v3. The columns of P are equal to [u1]c, [u2]c,
[u3]c. So:
=
Thus:
2 8 7 1 2 1
= 2
5
2 3 5 0
3
2
6
4 6 1
We can multiply t
CS132 Assignment #6
[email protected]
U92603848
April 24, 2017
Assignment #6
1. First, we need to find the eigenvalues of matrix A.
We must find 2 scalars (2x2 matrix) that satisfy the characteristic equation:
. 4 .3
0
.4
.3
=
=
. 4 1.2
0
.4
1.2
To fi
Name: Kevin Xia
User ID: U92603848
Email: [email protected]
Date: 2/12/2017
Assignment 2
1) Exercise 1
a) Notice that line M is parallel to the vector qp:
2
=
5
3
=
1
3
2 5
=
=
1 5
6
Since line M is essentially vector p with some direction vector, the param
Kevin Xia
U92603848
[email protected]
2/21/17
1) Consider two vectors a and b. T is a linear transformation if the following conditions hold:
( + ) = () + ()
() = ()
sad
For all a and b in the domain in T and for all scalars c.
So, assume vector a is:
a1
=
CS 132 at Boston University
Assignment3
Due 21st February @ 11:59pm using gSubmitonly.
1. Let T: IR3 IR3 be a transformation that projects each vector x = (x1 , x2 , x3 ) onto the plane x2 = 0
so T (x) = (x1 , 0, x3 ). Show that T is a linear transformati
CS132 Homework #2 (40 pts)
1. (30 pts) For numbers 15 from HW1, do the following
a. write a coefficient matrix,
b. an augmented matrix (with a righthand side),
c. compute an echelon form of an augmented matrix (show your intermediate steps and
the resul
CS132 Homework #3 (8 pts)
1) (2 pts) How long would it take for a PC computer with a performance of 50 gigaflops (giga = 109) to
solve a linear system of 2 million unknowns and equations via Gaussian elimination? How long
would it take with LU method, ass
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 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