Prove that a projection Pe(x) is a linear operator on a Hilbert space, ,H and the norm of projection Pe(x) =1 except for trivial case when e = {0}.

1.) Find a house: House Price is ($350,000)

2.) Use interest rate of: 6% compounded monthly for a 30 year loan and 5.5% compounded monthly for a 15 year loan.

3.) Determine how much a monthly principal and interest payment on your house would be if you financed it:

a. for 30 years

b. for 15 years

c. for 30 years with 20% down

d. for 15 years with 10% down

4.) How much would you pay for the home over the length of the loan under each scenario? How much of this is interest?

5.) A mortgage payment is made up of principal and interest payments from your loan as well as taxes and insurance payments. If the amount for taxes and insurance doubles your loan payment, how much would your mortgage payment be under each of the four scenarios?

6.) Which option do you feel is best? Why?

Prove that every nontrivial finite group has a composition series

Let 2a > -1, If the area of the region of the plane defined by {(x,y) : x ≥ 0, 2y-x ≥ 0, ax+y-3 ≤ 0} is equal to 3, then the value of a, lies in
Answer key: (0.75, 1.5)

9.2 Give 3 examples of equivalence relations and describe the equivalence classes.

9.3 Let R be an equivalence relation on a set S. Prove that two equivalence classes are either equal or do not intersect. Conclude that S is a disjoint union of all equivalence classes.

Describe various spaces associated with an m × n matrix A, such as null space, row space. column space and eigenspace. What are the relationships among them? How does the concept of a linear transformation and its properties relate to matrices and those spaces of the matrices?

Let V be a Hilbert space. Let f(x) = ∥x∥ for x ∈ V. Using the definition of Frechet differentiation, show that ∇f(x) = x for all x ̸= 0. Furthermore, show that f(x) is not Frechet differentiable at x = 0.

17. Integrate the function sin(x) from 0 to π using Riemann and Simpson integration methods with N subintervals, where N increases by factors of 2 from 2 to 256 (i.e. N = 2, N = 4, etc.). Plot the relative error ∆I/I = [| estimated value - true value |/(true value)], assuming the

highest value of N gives the ‘true’ value. Which method converges the fastest?

Solve the recurrence equations by Substitution

a) T(n) = 4T (n/2) + n, T (1) = 1

b) T(n) = 4T (n/2) + n² , T (1) = 1

c) T(n) = 4T (n/2) + n³ , T (1) = 1

A telephone sales force can model its contact with customers as a Markov chain. The six states of the chain are as follows:
State 1 Sale completed during most recent call
State 2 Sale lost during most recent call
State 3 New customer with no history
State 4 During most recent call, customer’s interest level low State 5 During most recent call, customer’s interest level medium State 6 During most recent call, customer’s interest level high
Based on past phone calls, the following transition matrix has been estimated:
1100000 20 1 0 0 0 0 3 0.10 0.30 0 0.25 0.20 0.15
P  4 0.05 0.45 0 0.20 0.20 0.10
5 0.15 0.10 0 0.15 0.25 0.35
6 0.20 0.05 0 0.15 0.30 0.30 
a) For a new customer, determine the average number of calls made before the customer buys the product or the sale is lost.
b) What fraction of new customers will buy the product?
c) What fraction of customers currently having a high degree of interest will buy the
product?
d) Suppose a call costs 20 TL and a sale earns 200 TL in revenue. Determine the “value”
of each type of customer.

Discrete math

1) Using MatLab (any language is fine just say what language is used).

function d=lemma_gcd(a,b)
%This program will return the greatest common divisor for input variables a
%and b where a and b are integers such that they are not both 0. It uses
%Lemma 4.8.3 on page 225.

%Note: gcd(a,b)=gcd(-a,b)=gcd(a,-b)=gcd(-a,-b), so
a=abs(a);
b=abs(b);

%This following section sets up the arrays we will use to store the
%changing values of our variables as a sequence. "c" will be the count
%variabe, stored in "C", "a" and "b" will also bechanging and stored in "A"
%and "B" arrays through the iterations of the loops.

c=0;
C(1)=c; %Stores c=0 in the array, C
%Next find the initial a and b to kick the program into gear

if a==0 & b==0
fprintf('You cannot input these variables\n')
d=inf;
else
if floor(a)~=a
fprintf('You cannot input these variables\n')
d=inf;
if floor(b)~=b
fprintf('You cannot input these variables\n')
d=inf;
  
  
  
  
  
  
  
  
end
end
end