Determine whether the set with the definition of addition of vectors and scalar multiplication is a vector space. If it is, demonstrate algebraically that it satisfies the 8 vector axioms. If it's not, identify and show algebraically every axioms which is violated. Assume the usual addition and scalar multiplication if it's not defined. V = { (x1, x2, x3) ∈ R^3 | x1 > or equal to 0, x2 > or equal to 0, x3 > or equal to 0}

Determine whether the set with the definition of addition of vectors and scalar multiplication is a vector space. If it is, demonstrate algebraically that it satisfies the 8 vector axioms. If it's not, identify and show algebraically every axioms which is violated. Assume the usual addition and scalar multiplication if it's not identified. V = R^2 , < X1 , X2 > + < Y1 , Y2 > = < X1 + Y1 , 0> c< X1 , X2 > = < cX1 , cX2 >

Determine whether the set with the definition of addition of vectors and scalar multiplication is a vector space. If it is, demonstrate algebraically that it satisfies the 8 vector axioms. If it's not, identify and show algebraically every axioms which is violated. Assume the usual addition and scalar multiplication if it's not defined. V = {all polynomials with real coefficients with degree > or equal to 3 and the zero polynomial}

In the real projective plane, list all possible + / - (positive and negative) patterns for three coordinates. Then match these triples to regions of the Fundamental Triangle. Also, describe the location of the usual four Euclidean quadrants in the Fundamental Triangle.

1. Use the method of exhaustion to prove the following statement: “For every prime number p between 30 and 58, 10 does not divide p − 9.”

2. Prove that 0.17461461 . . . is rational (digits 461 in the fractional part are periodically repeated forever).

Solve d) and e). I have provided answer for c) too required in e)

Winkler Furniture manufactures two different types of china cabinets: a French provincial model and a Danish Modern model. Each cabinet produced must go through three departments: carpentry, painting, and finishing. The table below contains all relevant information concerning production times per cabinet produced and production capacities for each operation per day, along with net revenue per unit produced. The firm has a contract with an Indiana distributor to produce a minimum of 300 of each cabinet per week (or 60 cabinets per day). Owner Bob Winkler would like to determine a product mix to maximize his daily revenue.

Formulate as an LP problem and obtain the revenue

c. What is the total Revenue at the optimal solution?

d. Bob Winkler wants to add this requirement to his production policy: To produce at least as many French Provencial cabinets as Danish Modern. How many French Provencial and Danish Modern Cabinets should Bob produce?

e. What is the impact on revenue of the solution in part d compared to the result in part c?

Solution for c)

Optimal Solution:

X1 = 60 X2 = 90

Revenue

Revenue =28*60+25*90= $3,930

#1 (a) Express, if possible,b= (7,8,9) as a linear combination of u1= (2,1,4),u2= (1,−1,3),u3= (3,2,5)

(b) What is Span{u1,u2,u3}?

(c) Is the set{u1,u2,u3}linearly independent?

#2 Considerv1= (1,2,1,1),v2= (1,1,3,1) and v3= (3,5,5,3). Find a homogeneoussystem the solution of which is Span{v1,v2,v3}. (Hint: Consider x= (x1, x2, x3, x4) where x=sv1+tv2+rv3 and look for conditions on x1, . . . , x4 where this system is consistent)

3.2.1. Find the Fourier series of the following functions:

(g) | sin x |

(h) x cos x.

Let V = {P(x) ∈ P10(R) : P'(−4) = 0 and P''(2) = 0}. If V= M_3×n(R), find n.

Recall P2(t) is the set of polynomials of order less than or equal to 2. Consider the the set of vectors in P2(t).

B={t^2,(t−1)^2,(t+1)^2}

(a) Show B is a basis for P2(t).

(b) If E={1,t,t^2}is the standard basis, calculate the change of basis matrices PE→B and PB→E

(c) Given v= 2t^2−5t+ 3, find its components in B

Given a sequence of closed intervals arranged so that each interval is a subinterval of the one preceding it and so that the lengths of the intervals shrink to zero, then there is exactly one point that belongs to every interval of the sequence. (This is known as the nested interval property)

3. The California Emmisons cap was set at 322 millon metric tons of carbon dioxide equivilent in 2016 and was expected to drop to 303 million metric tons of arbon dioxide equivilent in 2024 if we assume that the decreases in the emmisons cap is linearly each year, then do the followig thre parts:

a. Determine the linear quation for the amount of emmisions, e (in millions of metric tons), in terms of the number of years after 2000, t. Show the steps that you performed to arrive at your answer. Your final answer should be in the form of a short summary sentence. Place a box around your final answer.

B. Interpret the slope of the equation in the context of the problem using the idea of the steepnes property.

C. If this trend continues, then when will there be approximately 290 million metric tons of carbon dioxide emmited. Show, in detail how you arrived at your answer. Write a short summary as your final answer.

most important part c)

Determining whether a quantified logical statement is true and translating into English, part 2.

infoAbout

In the following question, the domain of discourse is a set of male patients in a clinical study. Define the following predicates:

• P(x): x was given the placebo
• D(x): x was given the medication
• A(x): x had fainting spells
• M(x): x had migraines

Suppose that there are five patients who participated in the study. The table below shows the names of the patients and the truth value for each patient and each predicate:

For each of the following quantified statements, indicate whether the statement is a proposition. If the statement is a proposition, give its truth value and translate the expression into English.

(c) ∃x M(x) ∧ D(x)

(f) ∀x ((M(x) ∧ A(x)) → ¬D(x))

(g) ∃x (D(x) ∧ ¬A(x) ∧ ¬M(x))

(h) ∀x (D(x) → (A(x) ∨ M(x)))

LU Decomposition

(i). Prove that for n equal to 2 or 3 there is a non-singular square (n by n) matrix which has no LU decomposition with L unit lower triangular and U upper triangular. (In fact, this is true for any integer ≥ 2.)

(ii). We will see that all non-singular square matrices do have an LUP decomposition (some time soon in class). Here P is a permutation matrix, also defined in Appendix D and used in Chapter 28.

Show that the inverse of a permutation matrix P is also a permutation matrix. (You can do this by explicitly defining what P −1 when P is given.)