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.

An important practice is to check the validity of any data set that you analyze. One goal is to detect typos in the data, and another would be to detect faulty measurements. Recall that outliers are observations with values outside the “normal” range of values of the rest of the observations.

Specify a large population that you might want to study and describe the type numeric measurement that you will collect (examples: a count of things, the height of people, a score on a survey, the weight of something). What would you do if you found a couple outliers in a sample of size 100? What would you do if you found two values that were twice as big as the next highest value?

You may use examples from your area of interest, such as monthly sales levels of a product, file transfer times to different computer on a network, characteristics of people (height, time to run the 100 meter dash, statistics grades, etc.), trading volume on a stock exchange, or other such things.

Let F be a field, and recall the notion of the characteristic of a ring; the characteristic of a field is either 0 or a prime integer.

Show that F has characteristic 0 if and only if it contains a copy of rationals and then F has characteristic p if and only if it contains a copy of the field Z/pZ.

Show that (in both cases) this determines the smallest subfield of F.

A machine in a factory has an error rate of 10 parts per 100. The machine normally runs 24 hours a day and produces 30 parts per hour. Yesterday the machine was shut down for 4

1. Develop an assembly language program for 89c51 to do following {Marks 10}

1. Store any 8 values of one byte each anywhere in Scratch Pad area of RAM. You are bound to use loop to store values.
2. Find Mean of these 8 values (use shift operator for division) and send it to port P2 (formula for to find mean is given below)
3. Find the lowest number among the numbers saved in part (1), take its 2’s complement and send it to P3

Formula for mean

μ=i=1nXin

Where X is values, and n is total number of values

1. Develop an assembly language program for 89c51 to do following

1. Store any 8 values of one byte each anywhere in Scratch Pad area of RAM. You are bound to use loop to store values.
2. Find Mean of these 8 values (use shift operator for division) and send it to port P2 (formula for to find mean is given below)
3. Find the lowest number among the numbers saved in part (1), take its 2’s complement and send it to P3

Formula for mean

μ=i=1nXin

Where X is values, and n is total number of values