A certain bookstore chain has two stores, one in San Francisco and one in Los Angeles. It stocks three kinds of book: hardcover, softcover, and plastic (for infants). At the beginning of January, the central computer showed the following books in stock.

Suppose its sales in January were as follows: 200 softcover, 400 hardcover, and 400 plastic books sold in Los Angeles, and 1,300 softcover books, 1,900 plastic books, and 600 hardcover books sold in San Francisco.

(a) Write the given inventory table as a 2 ✕ 3 matrix.

(b) Write the given sales figures as a similar matrix.

(c) Compute the inventory remaining in each store at the end of January.

1. (Covering part 1 of the lecture slides)

Consider the subset A of natural numbers, defined by A={ all natural numbers n : n is congruent to 1 (mod 3) }

First consider G: N -------> {0,1} to be the characteristic function of A.

a) Determine G(23) and G(25)

b) determine which sequence of 0,1 is this set associated with. (Write about 9 terms of it, or describe it with a formula.)

c) Which real number in the interval (0,1) in base 2 is associated with this set?  

Hi, can you please verify with the basic code of a beginner MATLAB user, and try explaining how to approach if possible. Thanks a lot! :)

1. Answer the following questions regarding complex numbers. You must provide all handwritten working and MATLAB code/outputs for the problems below.

(a) Express the complex number w = √ 3+i in complex exponential form. (Provide a handwritten solution)

(b) (1 mark) Use MATLAB to check your answer for part (a) by calculating the length and argument of w. (Provide your MATLAB code/outputs)

(c) Find all solutions to z^5 = √ 3 + i. (Provide a handwritten solution)

(d) Use MATLAB to plot the solutions found above. The solutions must be plotted on the same figure. (Provide you MATLAB code/outputs) (e) What do you notice about the solutions in the plot? (Provide a handwritten solution)

2. Let A = {1,2,3,4}. Let F be the set of all functions from A to A. Recall that IA ∈ F is the identity function on A given by IA(x) = x for all x ∈ A. Consider the function E : F → A given by E(f) = f(1) for all f ∈ F.
(a) Is the function E one-to-one? Prove your answer.

(b) Is the function E onto? Prove your answer.

(c) How many functions f ∈ F are there so that E(f) = E(IA)? Explain.

(d) How many onto functions f ∈ F are there so that E(f) = E(IA)? Explain.

A mass that weight 5lb stretches a spring 3in. The system is acted on by an external force 6sin⁡(8sqrt2 t)lb. If the mass is pulled down 4in and then released, determine the position of the mass at any time t. Use 32ft/s2 as the acceleration due to gravity. Pay close attention to the units.

u(t)=

Let U and V be vector spaces, and let L(V,U) be the set of all linear transformations from V to U. Let T_1 and T_2 be in L(V,U),v be in V, and x a real number. Define vector addition in L(V,U) by (T_1+T_2)(v)=T_1(v)+T_2(v) , and define scalar multiplication of linear maps as (xT)(v)=xT(v). Show that under these operations, L(V,U) is a vector space.

Let f(x) = sin(πx).

• x0 = 1,x1 = 1.25, and x2 = 1.6 are given. Construct Newton’s DividedDifference polynomial of degree at most two.

• x0 = 1,x1 = 1.25,x2 = 1.6 and x3 = 2 are given. Construct Newton’s Divided-Difference polynomial of degree at most three.

1. Let A be an m x n matrix. Prove that Ax = b has at least one solution for any b if and only if A has linearly independent rows.

2. Let V be a vector space with dimension 3, and let V = span(u, v, w). Prove that u, v, w are linearly independent (in other words, you are being asked to show that u, v, w form a basis for V)

Select a product or service of interest, and develop a segmentation scheme chart with 16 segments for the market, that involves at least 4 variables. Select and justify the choice of a target market.

Construct a conformal equivalence between a “half-strip” S1 := {z : 0 < Im z < 1,Re z > 0} and a “full strip”

S2 := {z : 0 < Im z < 1}

Let A and B be orthogonal Latin squares of order n, with symbols 0, 1 …, n – 1. Let B’ be obtained from B by permuting the symbols in B. Show that A and B’ are still orthogonal.

Let Dn be the set of positive integers that divide evenly into n. List the elements of each of the sets D₆, D₁₆, D₁₂, and D₃₀

Let f(x,y) = 3x³ + 3x² y − y³ − 15x.

a) Find and classify the critical points of f. Use any method taught during the course (the second-derivative test or completing the square).

b) One of the critical points is (a,b) = (1,1). Write down the second-degree Taylor approximation of f about this point and motivate, both with computations and with words, how one can see from this approximation what kind of critical point (1,1) is. Use completing the square.

Answer for a and be should be answered independently.

Let (X,d) be a metric space, and

a) let A ⊆ X. Let U be the set of isolated points of A. Prove that U is relatively open in A.

b) let U and V be subsets of X. Prove that if U is both open and closed, and V is both open and closed, then U ∩ V is also both open and closed.