COT 5615/ CIS 4930
Homework 1
1. Let N be the set of natural numbers. Using Peano axioms,
(a) Prove the cancellation law of addition, i.e., if m, n, p N and m + p = n + p, then m = n.
(b) Prove that m
61
HOMEWORK SOLUTIONS G
Math Review Solutions.
Linear Algebra Problem Set 2.
1. The eigenvalues and eigenvectors satisfy the equation Avk = k vk for k = 1, 2. This
is equivalent to (A k I) vk = 0. Dro
58
HOMEWORK SOLUTIONS G
Math Review Solutions.
Linear Algebra Problem Set 1.
1. C = AB =
7 10
3
14 .
16
32
T
T
2. Since [2, 3, 2] = [2, 3, 2] , they are linearly dependent. Hence, they are not
a basis
HOMEWORK SOLUTIONS G
67
Math Review Solutions.
Probability Problem Set 1.
1. Recall that the Borel -algebra on the interval (0, 1) is the smallest -algebra containing all the countable unions and inte
30
MATHEMATICS REVIEW G
A.1.1
Matrices and Vectors
Definition of Matrix. An M xN matrix A is a two-dimensional array of numbers
a11 a12 . . . a1N
A=
a21
a22
.
.
.
.
.
.
aM 1
aM 2
.
.
a2N
.
.
aM N
58
HOMEWORK PROBLEMS
Math Review Homework.
Probability Problem Set 1.
1. -algebras
(a) The Fisher Iris Date Set is a famous data set in Pattern Recognition and Machine
Learning. It can be found in man
54
HOMEWORK PROBLEMS
Math Review Homework.
Linear Algebra Problem Set 2.
Let A and B be the following matrices:
5
2
32
23
5
2
"
A=
#
=
1
2
5
3
9
3
2
and B =
5
3
2
3
2
11
2
"
=
1
2
9
#
3
3
11
1. F
Homework 1
(due Thursday, September 3rd, 2015)
I: Linear Systems
1. Give a simple example of a 3 3 matrix A of rank 2 and demonstrate that a non-zero vector z exists
for which Az = 0.
2. The condition
Homework 3
(due Tuesday, September 29th, 2015)
I: The Groups of StrangeBrew
There are n odd citizens living in StrangeBrew. Their main occupation was forming various groups (with
Were strange doesnt m
30
APPENDIX: MATHEMATICS REVIEW G
12.1.1
Matrices and Vectors
Definition of Matrix. An M xN matrix A is a two-dimensional array of numbers
2
6
6
A=6
4
a11
a21
a12
a22
.
.
.
.
.
.
.
aM 1
aM 2
.
.
a1N 3
LINEAR ALGEBRA
39
Chapter 12 Exercises. Linear Algebra.
1. Let A and B be the following matrices:
1
A= 3
4
2
5
2
2 and B =
6 4
6
Compute the product C = AB.
2. Determine mathematically whether the set
COT 5615/ CIS 4930
Homework 2
1.
Let X be an infinte set. For p X and q X , define
(
1,
d (p, q) =
0,
(if p 6= q) ,
(if p = q) .
Prove that this is a metric. Which subsets of the resulting metric spac
COT5615 Mathematics for Intelligent Systems Fall 2014
Home Work Assignment 2: Due Wednesday 10/08/14 before class
1. Recall the following definitions from class.
Defn 1 A function f : X Y for metric s
COT 5615/ CIS 4930
Homework 2
1.
Let X be an infinte set. For p X and q X , define
(
1,
d (p, q) =
0,
(if p 6= q) ,
(if p = q) .
Prove that this is a metric. Which subsets of the resulting metric spac
COT 5615/ CIS 4930
Homework 1 solution
1.
Let N be the set of natural numbers. Using Peano axioms,
(a) Prove the cancellation law of addition, i.e., if m, n, p N and m + p = n + p, then m = n.
Ans.
We
COT5615 Mathematics for Intelligent Systems Fall 2014
Home Work Assignment 4: Due Friday 12/05/14 before class
1. Prove that if A is a square matrix (i.e., n n) then the eigen values of AT A and AAT a
COT5615 Mathematics for Intelligent Systems Fall 2014
Home Work Assignment 1: Due Monday 09/15/14 before class
1. Prove that
3 is not a rational number.
2. Recall that Rational numbers are defined as
COT5615 Mathematics for Intelligent Systems Fall 2012
Home Work Assignment 2: Due Friday 09/28/12 before class
1. Prove that the sequence, xn+1 =
xn
2
+
1
xn
with x0 = 1 is a Cauchy sequence.
2. Prove
COT5615 Mathematics for Intelligent Systems Fall 2011
Home Work Assignment 1: Due Monday 08/29/11 before class
1. Let p be any prime number. Prove that no natural numbers a, b, and c exist such that a
COT5615 Mathematics for Intelligent Systems Fall 2012
Home Work Assignment 1: Due Friday 09/07/12 before class
1. Prove that there exist no prime numbers p1 , p2 , p3 and p4 such that p1 p2 = p3 p4 .
COT5615 Mathematics for Intelligent Systems Fall 2014
Home Work Assignment 3: Due Monday 11/17/14 before class
In the problems below, the matrix
3 1 2
A = 4 7 11
0 1
9
1. Prove that for all vectors x
COT5615 Mathematics for Intelligent Systems Fall 2012
Home Work Assignment 3: Due Monday 10/29/12 before class
In the problems below, the matrix
3 1 2
A = 4 7 11
0 1
9
1. Prove that for all vectors x
COT5615 Mathematics for Intelligent Systems Fall 2012
Home Work Assignment 4: No submission reqiired
1. Prove that if A is a square matrix (i.e., n n) then the eigen values of AT A and AAT are the sam
38
APPENDIX: MATHEMATICS REVIEW G
12.1.4
Eigenanalysis and Symmetric Matrices
Eigenvalues, Eigenvectors, and Symmetric Matrices play a fundamental role in describing
properties of spectral data obtain