1. Let w,v1,...,vp ∈ Rn and suppose that w ∈ Span{v1,...,vp}. Show that Span{w,v1,...,vp} = Span{v1,...,vp}.

2. Let v1,...,vp,w1,...,wq ∈ Rm. Is the following statement True or False?

   “If {v1, . . . , vp} is linearly dependent then {v1,...,vp,w1,...,wq}

   is linearly dependent.”
   If you answer True, provide a complete proof; if you answer False, provide a counter-example. Linear Algebra. Please show both!

Discrete Math: Prove the following statements:

By Direct Proof:

i) For All Natural Numbers p q & r:   if p divides q and q divides r then p divides r.

ii) For All integers a & b: if a = b mod 12, then a = b mod 6;

(show that the converse is not true with an example [if a= b mod 6, then a = b mod12]))

iii)Show that for sets F & G

- If F is a subset of G , then F = F intersection G

- If F = F intersection G, then G = F union G

- If G = F union G, then F is a subset of G

Show that for Real numbers a & b : (a+b)²>=4ab when a &b >=0

Please Prove the following, be clear and percise.

By removing sets with ever decreasing length, show that we can construct a "Cantor-like" set which has positive length. How large can we make the length of this set?

Suppose you are given an undirected, connected, weighted graph G. You want to find the heaviest weight set of edges that you can remove from G so that G is still connected. Propose an optimal greedy algorithm for this problem and prove that it is optimal. (Hint: use matroid theory!).

In question 1, do not use any properties of odd and even numbers outside of
their definitions. However, for questions 2 and 3, you can use the following
parity results without having to prove them each time.
For any integers a and b, the product a.b is odd if and only if a and
b are both odd.
For any integers, a and b, the sum a + b is even if and only if a and b
have the same parity.
For all questions, let x; y 2 Z.
1. Prove that if 11x - y is even, then x and y have the same parity.
2. Prove that 5xy + x + y is odd if and only if x or y is odd.
3. Prove that if 2|3x² - 1, then 4 |x² + 7.

A shipping company ShipCo supplies four destinations (D1,D2,D3,D4) from four sources (S1,S2,S3,S4). The shipping cost (in dollars) per shipment from each source to each destination is given below.

The four sources make 10, 20, 20, and 10 shipments per month, respectively. The four destinations need to receive 20, 10, 10, and 20 shipments per month, respectively. The manager of ShipCo now wants to determine the best plan for how many shipments to send from each source to each destination each month. The objective is to minimize the total shipping cost. Answer the following questions:
(a) Formulate this transportation problem as an LP. Clearly explain your variables and constraints.
(b) Use the Northwest corner method to find an initial BFS.
(c) Starting from the initial BFS in part (b), apply the transportation algorithm to solve this problem.

Let A be an n x n matrix satisfying A²=A (idempotent). Find all eigenvalues and eigenvectors of A.

I know that the eigenvalues are 0 and 1 -- I do not know how to find the eigenvectors.

a) Which variables are nonbasic, which ones are basic, and what are their respective values?

b) If I were minimizing the objective, which nonbasic variables are legitimate candidates to enter the basis? How about if I were maximizing?

c) Suppose I decided to enter x4 into the basis and increase its value by 100 units. Without doing any pivoting, can you say what the new objective value will be? Explain clearly.

d) Corresponding to this 100 unit increase in x4 what are the adjustments that need to be made to the values of the current basic variables in order to maintain feasibility? Explain clearly.

e) What is the maximum amount of increase possible in the value of x4? Explain clearly.

f) Again, without doing any pivoting, can you say what the value of the objective will be after the next iteration is completed?

Suppose $22,000 is invested at an annual rate of 5% for 20 years. Find the future value if interest is compounded as follows

a. Annually

b.Quarterly

c.Monthly

d.Daily (365 days)

e.Continuously

Congratulations, you just won the lottery! In one option presented to you, you will be paid one million dollars a year for the next 25 years. You can deposit this money in an account that will earn 5% each year.

a. Let M(t) be the amount of money in the account (measured in millions of dollars) at time t (measured in years). Set up a differential equation that describes the rate of change in the amount of money in the account. Two factors cause the amount to grow – first, you are depositing one million dollars per year and second, you are earning 5% interest.

b. The second option presented to you is to take a lump sum of 10 million dollars, which you will deposit into a similar account. Set up a new initial value problem (that is, differential equation with initial condition) to model this situation.

c. At what time does the amount of money in the account under the first option overtake the amount of money in the account under the second option?

a.Prove that {12a+ 4b | a, b ∈ Z}={4c |c ∈ Z}.

(b) Prove that{20a+ 16b|a, b ∈ Z}={28m+ 32n|m, n ∈ Z}.

(c) Leta, b ∈ Z−{0}. Prove that{x ∈ Z |ab divides x}⊆{x ∈ Z |a divides x}.

(d) Prove that{16n|n∈Z}⊆{2n|n ∈ Z}.

A particle of mass m is in a force field described by the equation,

                                 F(x) = bx

Here b is a positive constant. At t = 0, the particle is released from the origin (x₀ = 0) with an initial velocity V₀

a) Build a differential equation, and use a 'trial solution' to find the position of the particle as a function of time, x(t)

b) Use the 'separating variables and integration' method to find x(t)

schematize the following passage:

Ketogenic diet is well known for rapid weight loss. The diet protocol restricts carbohydrate consumption to 30 gm per day for an 80 kg adult. In order to protect the lean body mass, the protein intake should fall between 1.2-1.7 gm per kg BW per day, less than that may cause a loss of lean muscles. The protein intake should be moderate as an excessive amount of protein is metabolized to convert into amino acids, which are required for gluconeogenesis high-fat diet results in the shift of metabolism, depleted levels of glucose and enhanced oxidation of fatty acids that form ketone bodies to provide energy. The use of electrolytes such as sodium and potassium is highly recommended for maintaining nitrogen balance with the preservation of the fuctional tissue in the body.

Samreen Aziz, and Hina Rehman. “Mechanism and Benefits of Ketogenic Diet for Weight Loss and Health.” Rewal Medical Journal, vol. 44, no. 4, 2019, pp. 880–882., http://www.rmj.org.pk/fulltext/27-1545295565.pdf?1583710925.

1. For the initial-value problem, ?'(?) = ?? sin(5?) , ?(0) = 0.5,
2. Use the Euler Method with step size of 0.1 to calculate the approximate solution for the

differential equation for the initial condition y(0) = 0.5 and plot the result on the direction

field graph you produced in paragraph 1. Then do the same for the results of using the Euler Method with step sizes of 0.01 and then 0.005.

3. For the initial-value problem in paragraph 2 above, ?'(?) = ?? sin(5?) , ?(0) = 0.5,

find the approximate the value of y when x = 4 using the Euler Method for each of the three different step sizes you used in paragraph 2.

Type or write:
“For the Euler Method with a step size of 0.1, y(4) =”, followed by the answer accurate

to five decimal places.
“For the Euler Method with a step size of 0.01, y(4) =”, followed by the answer accurate

to five decimal places.
“For the Euler Method with a step size of 0.005, y(4) =”, followed by the answer accurate

to five decimal places.

4. Use the fourth-order Runge-Kutta Method with step sizes of 0.1, 0.01, and 0.005 to calculate approximate solutions for the same initial-value problem as paragraphs 2 and 3 and plot these approximate solutions on the direction field graph.
   • Attach a direction field graph showing the fourth-order Runge-Kutta Method results using a step size of 0.1.
   • Attach a second direction field graph showing the fourth-order Runge-Kutta Method results using a step size of 0.01.
   • Attach a third direction field graph showing the fourth-order Runge-Kutta Method results using a step size of 0.005.
   • Type or write a brief explanation of what happens with the plots as the step size is reduced.
5. For the initial-value problem in paragraphs 2 and 3 above, find the approximate the value of y when x = 4 using the fourth-order Runge-Kutta Method for each of the three different step sizes you used in paragraph 4.

Type or write:
“For the fourth-order Runge-Kutta Method with a step size of 0.1, y(4) =”, followed by

the answer accurate to five decimal places.
“For the fourth-order Runge-Kutta Method with a step size of 0.01, y(4) =”, followed by

the answer accurate to five decimal places.
“For the fourth-order Runge-Kutta Method with a step size of 0.005, y(4) =”, followed by

the answer accurate to five decimal places.