Thank You

Define the gcd of three integers a, b, c as the largest common divisor of a, b, c, and denote it by (a, b, c). Show that (a, b, c) = ((a, b), c) and that (a, b, c) can be expressed as a linear combination of a, b, c.

a, The vectors v1 = < 0, 2, 1 >, v2 = < 1, 1, 1 > , v3 = < 1, 2, 3 > , v4 = < -2, -4, 2 > and v5 = < 3, -2, 2 > generate R^3 (you can assume this). Find a subset of {v1, v2, v3, v4, v5} that forms a basis for R^3.

b. v1 = < 1, 0, 0 > , v2 = < 1, 1, 0 > and v3 = < 1, 1, 1 > is a basis for R^3 (you can assume this.) Given an arbitrary vector w = < a, b, c > write w as a linear combination of v1, v2, v3.

c. Find the dimension of the space spanned by x, x-1, x^2 - 1 in P2 (R).

In parts a, b, and c, determine if the vectors form a basis for the given vector space. Show all algebraic steps to explain your answer.

a. < 1, 2, 3 > , < -2, 1, 4 > for R^3

b. < 1, 0, 1 > , < 0, 1, 1> , < 2, 0, 1 > for R^3

c. x + 1, x^2 + 1, x^2 + x + 1 for P2 (R).

Part a (worth 60 pts): Formulate a linear programming model (identify and define decision variables, objective function and constraints) that can be used to determine the amount (in pounds) of Brazilian Natural and Colombian Mild that will maximize the total contribution to profit. For “Part a” you do NOT need to solve this problem using Excel, you just need to do the LP formulation in the standard mathematical format.

Part b (bonus worth 20 pts): Solve the LP problem that you formulated in “Part a” using Excel. Give the values of each decision variable and the objective function. You MUST attach a copy of the solution report.

1.29 Prove or disprove that this is a vector space: the real-valued functions f of one real variable such that f(7) = 0.

Find all unlabeled trees on 2,3,4 and 5 nodes. How many labeled trees do you get from

each? Use this to find the number of labeled trees on 2,3,4 and 5 nodes.

Discuss the situation of a linear program that has one or more columns of the A matrix equal to zero. Consider both the case where the corresponding variables are required to be nonnegative and the case where some are free. (the available answer mentioned about the degenerate solution, which is still confusing, why and how the solution is degenerate if one or more columns of the A matrix equal to zero )

Let f: X-->Y and g: Y-->Z be arbitrary maps of sets

(a) Show that if f and g are injective then so is the composition g o f

(b) Show that if f and g are surjective then so is the composition g o f

(c) Show that if f and g are bijective then so is the composition g o f and (g o f)^-1 = g ^ -1 o f ^ -1

(d) Show that f: X-->Y is injective iff there exists h: Y-->X such that h o f = id sub x

(e) Show that f: X-->Y is surjective iff there exists h: Y-->X such that f o f = id sub y. The only if requires requires the axiom of choice.

Let

A, B

be sets and

f

:

A

→

B

and

g

:

B

→

C

. Characterize when

g

◦

f

:

A

→

C

is a bijection.

Show the conversion of the following numbers with the standard conversion procedure.

1. (AD)_H = ( ?)₂      = ( ? )₁₀

2. (24)₇   = ( ? )₁₀    = ( ? )₂

3. ( 10011001)₂      = ( ? )₈      = (?)₁₆

4. (334.2301)₁₀   = ( ? )₂

Using only the rules of SD, derive the following rules of SD+:

a) Disjunctive Syllogism (one case) :

P v Q
~P

b) Commutation: (wedge only) :

P v Q <--> Q v P

c) Implication :

P ⊃ Q <--> Q ⊃ ~P

d) Double Negation

P <--> ~~P

e) DeMorgan

(~(P & Q) <--> ~P v ~Q only)

Use SL sentences rather than metavariables in your derivations. Each direction of replacement rules must be shown (that is, you are proving "equivalence in SD" in these cases)

Your remodeled home will have an open floor plan with the dining area, living area, and kitchen in one large room. You will have a fireplace along one of the walls, and you plan to mount your TV above the fireplace. Your sofa will be directly across from the fireplace/TV, in the middle of the room. You want to have an electrical outlet installed on the floor so that it will be under the sofa. This will allow you to have end tables on each side of the sofa with lamps that can be plugged in the floor outlet, and you can also use the outlet to charge your devices. You need to tell the builder how far from the fireplace/TV you want the floor outlet to be installed. Some things to consider:

a. The mantel of your fireplace is 53 inches above the floor, and the bottom of the TV will be 5 inches above the mantel.

b. Your TV is 48 inches (which is the diagonal measure from corner to corner). The width is approximately 41.7 inches.

c. Your sofa's height from the floor to the top of the seat cushion is 18 inches. Its depth from front to back is 32 inches, and the depth of the seat cushion from the front to back is 22 inches.

d. The optimal angle of elevation from your eyes to the middle of a TV screen is 0 degrees, but that is impossible since you want the TV above the fireplace. You find online that neck strain can be avoided if the angle of elevation from your eyes to the middle of the screen is no more than 15 degrees.

Describe where you want the floor outlet installed. There's no single correct answer to this question: you could make a reasonable argument for many distances, based on the considerations above, as well as on your height and other things you might consider. To get FULL credit, you must include detailed diagrams, all computations, and verbal explanations that will support your answer.

Please explain in the simplest form. Thanks for assisting!

Find a p-Sylow subgroup for each of the given groups, and prime p:

a. In Z₂₄ a 2-sylow subgroup

b. In S₄ a 2-sylow subgroup

_c. In A₄ a 3-sylow subgroup

3. Given is the function f : Df → R with F(x1, x2, x3) = x 2 1 + 2x 2 2 + x 3 3 + x1 x3 − x2 + x2 √ x3 . (a) Determine the gradient of function F at the point x 0 = (x 0 1 , x0 2 , x0 3 ) = (8, 2, 4). (b) Determine the directional derivative of function F at the point x 0 in the direction given by vector r = (2, 1, 2)T . (c) Determine the total differential dF of function F and use it to compute approximately the absolute and relative error in the computation of F(x 0 ) when the independent variables are from the intervals x1 ∈ [7.8, 8.2], x2 ∈ [1.9, 2.1], x3 ∈ [3.9, 4.1]. (14 points) 4. (a) Determine all points satisfying the necessary conditions of the Lagrange multiplier method for a local extreme point of the function f(x, y) = x 2 + y 2 subject to the constraint x 2 + 2y 2 − 2 = 0 . (b) Using the sufficient conditions, check whether the point (x ∗ , y∗ ; λ ∗ ) = (− √ 2, 0; −1) is a local minimum or maximum point and give the corresponding function value.

Consider the problem   maximize   Z = 5 x1 + 3 x2 + 2 x3 + 4 x4       

subject to                      

5 x1 + x2 + x3 + 8 x4 = 10                      

2 x1 + 4 x2 + 3 x3 + 2 x4 = 10                                    

X j > 0, j=1,2,3,4

(a) Make the necessary row reductions to have the tableau ready for iteration 0. On this tableau identify the corresponding initial (artificial) basic feasible solution. Also, identify the initial entering and leaving variables.

(b) Following the result obtained in (a) solve by the Simplex method, using the Big-M method.

(c) Solve by the Two-Phase method.