4. Verify that the Cartesian product V × W of two vector spaces V and W over (the same field) F can be endowed with a vector space structure over F, namely, (v, w) + (v ′ , w′ ) := (v + v ′ , w + w ′ ) and c · (v, w) := (cv, cw) for all c ∈ F, v, v′ ∈ V , and w, w′ ∈ W. This “product” vector space (V × W, +, ·) is commonly (and more appropriately) denoted as V ⊕ W, called the direct sum of V and W. The Euclidean plane R 2 ≡ R × R is in fact R ⊕ R. (Remark. This notion of direct sum can be extended to the direct sum of finitely many vector spaces V1 ⊕ V2 ⊕ · · · ⊕ Vk in a straightforward way.)

Consider the Newton-Raphson method for finding root of a nonlinear function
??+1=??−?(??)?′(??), ?≥0.
a) Prove that if ? is simple zero of ?(?), then the N-R iteration has quadratic convergence.
b) Prove that if ? is zero of multiplicity ? , then the N-R iteration has only linear convergence.

(6) In each part below, find a basis for R4 CONTAINED IN the given set, or explain why that is not possible: Do not use dimension
(a) {(1,1,0,0),(1,0,1,0),(0,1,1,0)}
(b) {(1,−1,0,0),(1,0,−1,0),(0,1,−1,0),(0,0,0,1)}
(c) {(1,1,0,0),(1,−1,0,0),(0,1,−1,0),(0,0,1,−1)}
(d) {(1,1,1,1),(1,2,3,4),(1,4,9,16),(1,8,27,64),(1,16,81,256)}

Prove that the entropy of a node never increases after splitting it into smaller children nodes.

Problem 1

a) Bernard operates a chocolate shop in Paris. The annual demand for chocolatecovered cherries is 2,500 units. The setup cost is $15 per order and the unit cost is $0.6. The holding cost per unit per year is 25% of the unit cost. Backordering is allowed at a cost of $0.6/unit/year. What is the optimum number of units per order and the corresponding backorder level? What is the expected number of orders per year? Assuming a 250 day working year, what is the expected time between orders? What are the total annual inventory costs? If delivery of the chocolates takes 2 days, at what level of stock should a new order be placed?

b) Suppose now that Bernard is offered the following whole unit discount model: If Bernard buys 400 units or less, it will cost him $0.7/unit, and if he buys more than 400 units it will cost him $0.5/unit. Will Bernard buys accept this offer?

It was estimated that in 2013, the Greenland Ice Sheet and its outlying ice caps were losing mass at a rate of about -393 Gt/y (gigatonnes per year).1 Suppose that in 2023, plans are put into place to gradually improve the rate of polar ice melt over the next 10 years, so that by 2033, the rate of ice melt in Greenland will be -93 Gt/y. Assuming that there are no further interventions after 2033, graph a function that represents the rate of ice melt from 2013 to 2043. Based on this graph, graph a function that could represent the total amount of ice in Greenland from 2013 to 2043.

1. Prove ├(∀x)A ∨ (∀x)B → (∀x)(A ∨ B).

2. Use the ∃ elimination technique —and ping-pong if/where needed— to show ├(∃x)A → (∃x)(B → A).

Appreciate you.

(2) More inclusion-exclusion counting: How many bit strings of length 15 have bits 1, 2, and 3 equal to 101, or have bits 12, 13, 14, and 15 equal to 1001 or have bits 3, 4, 5, and 6 equal to 1010? (Number bits from left to right. In other words, bit #1 is the left most bit and bit #15 is the right most bit.) Hint: The fact that the third bit appears in two of the required patterns means some special care will be needed to get the count correct.

For which values of λ does the system of equations

(λ − 2)x + y = 0

x + (λ − 2)y = 0

have nontrivial solutions? (That is, solutions other than x = y = 0.) For each such λ find a nontrivial solution.

Given f(x) = sin^2(x) and g(x) = sin^4(x) (give exact answers for all parts): a) Plot the functions on the x-interval [0, π]. Find the volume when the region enclosed by the curve and the x-axis is rotated about the line x = π. b) Find the area of the region. 1?b Aa and y = 1? b 1(f(x)2 −g(x)2)dx. Find the x-coordinate of the center of Aa2 mass of the region. In a print statement, explain why this answer makes sense based on the graph in part a). d) When the region rotates about the line x = π, how far does the center of mass travel? Multiply this value by the area. What do you notice when you compare your answer to part a)?

Python

Define error function and complementary error function, mathematically. Explain how these functions can be tabulated.

For each system described below say as much as possible about each system’s solution
set. Note which theorems you are using to reach your conclusions.
(a) A consistent system of 8 equations in 5 variables.
(b) A consistent system of 5 equations in 8 variables.
(c) A system of 4 equations in 9 variables.
(d) A system with 15 equations in 35 variables.
(e) A system with 8 equations and 5 variables. The reduced row-echelon form of the
augmented matrix of the system has 6 pivot columns.

Use LU decomposition to solve the following system of equations (show your work). Do not use a pivoting strategy, and check your results by using the matrix inverse to show that [A][A]-1= [I].

8x+ 2y−z=10

- 2x+4y+z=5

3x−y+ 6z=7

Using a power series methodology, obtain the general solution (form u = c1u1 + c2u2 + f(x)) to the equation u” + 4u = x.

2. Consider a homogeneous system of linear equations with m equations and n variables.

   1. (i) Prove that this system is consistent.

   2. (ii) Prove that if m < n then the system has infinitely many solutions. Hint: Use r (the number of pivot columns) of the augmented matrix.