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 multiplication is commutative, i.e., n m = m n, m, n N.
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 number of a matrix A is dened as
cond(A) = A A1 .
(1)
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 space are open? Which are closed? Which are compact?
Ans.
I
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 spaces X, Y is continuous at f (x) = y if for every
conv
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 same.
Q
2. Prove that if A is an n n real symmetric matrix
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 and y of an inner product space it is true that |hx, y
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 and y of an inner product space it is true that |hx, y
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 .
Your proof should only use facts that you consider elem
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, b, c < p and
p c = a b.
Your proof should only use fa
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 that the sequence, xn+1 =
xn
2
1
xn
with x0 = 1 is not
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 the quotient space (Z (Z\cfw_0)/ where the equivalence
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 are the same.
Q
2. Prove that if A is an n n real symmet
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 proceed by induction on p.
Induction base case: For ar
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 space are open? Which are closed? Which are compact?
2.
Sho
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 mean we are strangers being their motto), which at some