1. Consider a= 〈−3,1,−2〉, b= 〈−2,0,−1〉 and c= 〈−5,4,−3〉. Find the angles between the following vectors

2. Consider a=4i−j+5k, b=−i+4k and c=i−k Find the following scalar and vector projections

A database uses 20-character strings as record identifiers. The valid characters in these strings are upper-case letters in the English alphabet and decimal digits. (Recall there are 26 letters in the English alphabet and 10 decimal digits.) How many valid record identifiers are possible if a valid record identifier must meet all of the following criteria:

• Letter(s) from the set {A,E,I,O,U} occur in exactly three positions of the string.

• The last three characters in the string are distinct decimal digits that do not appear elsewhere in the string.

• The remaining characters of the string may be filled with any of the remaining letters or decimal digits.

1. (20 pts) For each of the following statements, please circle T (True) or F (False). You do not need to justify your answer. (a) T or F? Any eigenvector of a matrix is in the column space of the matrix. (b) T or F? The number of singular values of a matrix is also its rank. (c) T or F? If A is an m × n with m < n, then the dimension of its column space is greater than the dimension of its row space. (d) T or F? A symmetric matrix is diagonalizable. (e) T or F? The null space null(A) of a matrix A is orthogonal to the column space of AT . (f) T or F? Zero can be the eigenvalue of an elementary matrix. (g) T or F? If W is a vector space spanned by 4 vectors, them the dimension of W is 4.

(a) Show that there are, up to isomorphism, exactly 8 matroids whose underlying set has three elements. Calling the elements a,b,c, exhibit, for each of these matroids, its bases, cycles and independent sets.

(b) Consider the matroid M on the set E = {a,b,c,d}, where the bases are the subsets of E having precisely two elements. Detrmine all the cycles of M, and show that there is no graph G such that M is the cycle matroid M(G).

Determine if the following set forms a subspace in R^2.

The set is (x1,x2)^t ,in other words the column vector [x1,x2]. Can you go through each axiom and show your work?, I have a lot of difficulty with these types of questions and I want to make sure I understand.Thank you in advance.

(The “conjugation rewrite lemma”.) Let σ and τ be permutations.

1. (a) Show that if σ maps x to y then σ^τ maps τ(x) to τ(y).

2. (b) Suppose that σ is a product of disjoint cycles. Show that σ^τ has the same cycle structure as

   σ; indeed, wherever (... x y ...) occurs in σ, (... τ(x) τ(y) ...) occurs in σ^τ.

Write the following complex numbers in polar form, as ??^??.

1/3+4i

Derive the variation of parameters formula for the solution of the initial value problem for a non-homogeneous, linear system of first order, ordinary differential equations in terms of a fundamental matrix of solutions of the corresponding homogeneous problem.

Solve the given system of differential equations by systematic elimination.

(x(t), y(t)) =

Solve the differential equation by variation of parameters.

y'' + 3y' + 2y =

y(x) =

Problem 3. Throughout this problem, we fix a matrix A ∈ Fn,n with the property that A = A∗. (If F = R, then A is called symmetric. If F = C, then A is called Hermitian.) For u, v ∈ Fn,1, define [u, v] = v∗ Au. (a) Let Show that K is a subspace of Fn,1. K:={u∈Fn,1 :[u,v]=0forallv∈Fn,1}. (b) Suppose X is a subspace of Fn,1 with the property that [v,v] > 0 for all nonzero v ∈ X. (1) Show that [−, −] defines an inner product on X. (c) Suppose Y is a subspace of Fn,1 with the property that [v,v] < 0 for all nonzero v ∈ Y. (2) If X is a subspace with property (1), prove that X + K + Y is a direct sum, where K is defined in part (a).

Ruff, Inc. makes dog food out of chicken and grain. Chicken has 10 grams of protein and 5 grams of fat per ounce, and grain has 2 grams of protein and 2 grams of fat per ounce. A bag of dog food must contain at least 220 grams of protein and at least 180 grams of fat. If chicken costs 11¢ per ounce and grain costs 1¢ per ounce, how many ounces of each should Ruff use in each bag of dog food to minimize cost? HINT [See Example 4.] (If an answer does not exist, enter DNE.)

Linear Programming Problem 1:

George's Woodcarving Company manufactures two types of wooden toys: soldiers and trains. A soldier sells for $27 and uses $10 worth of raw materials. Each soldier manufactured increases George's variable labor and overhead costs by $14. A train sells for $21 and uses $9 worth of raw materials. Each Train built increases George's variable labor and overhead costs by $10. The manufacture of wooden soldiers and trains requires two types of skilled labor: carpentry and finishing. A soldier requires 3 hours of carpentry labor and 2 hours of finishing labor. A train requires 4 hours of carpentry labor and 1 hour of finishing labor. Each week, George's can obtain all the needed raw material but only 240 carpentry hours and 100 finishing hours. Demand for trains is unlimited, but at most 28 soldiers are bought each week. George wishes to maximize weekly profit (revenue – costs). The company wants to find out the optimal production strategy that maximizes the weekly profit.

First solve this problem graphically or using the Solver. Have the solved graph or spreadsheet ready. For graphical approach, you need to solve for the optimal solution by solving simultaneous equations after graphing.

Then answer the quiz questions.

1. How many decision variables are in this problem?

2. How many finishing hours are available in this problem?

3. What is the unit profit of a toy soldier? $____.

4. To produce 5 toy soldiers and 5 toy trains, how many carpentry hours are required?

5. To produce 5 toy soldiers and 10 toy trains, how many finishing hours are required?

6. In the optimal solution, how many toy soldiers are produced?

7. In the optimal solution, how many toy trains are produced?

8. What is the maximum total profit?

9. In the optimal solution, how many hours of carpentry labor in total are used?

10. In the optimal solution, how many hours of finishing labor in total are unused?

Let f and g be two functions whose first and second order derivative functions are continuous, all defined on R. What assumptions on f and g guarantee that the composite function f ◦g is concave?

1.  Inventory Management

Williams & Sons last year reported sales of $145 million, cost of goods sold (COGS) of $120 and an inventory turnover ratio of 5. The company is now adopting a new inventory system. If the new system is able to reduce the firm's inventory level and increase the firm's inventory turnover ratio to 8 while maintaining the same level of sales and COGS, how much cash will be freed up? Do not round intermediate calculations. Round your answer to the nearest dollar.

ANSWER: $  

2. Receivables Investment

Medwig Corporation has a DSO of 37 days. The company averages $4,250 in sales each day (all customers take credit). What is the company's average accounts receivable? Round your answer to the nearest dollar.

ANSWER: $

3. Cost of Trade Credit

What are the nominal and effective costs of trade credit under the credit terms of 1/20, net 30? Assume 365 days in a year for your calculations. Round your answers to two decimal places. Do not round intermediate calculations.

ANSWER: (percentages)

Nominal cost of trade credit:

Effective cost of trade credit:

4. Cost of Trade Credit

A large retailer obtains merchandise under the credit terms of 3/20, net 30, but routinely takes 65 days to pay its bills. (Because the retailer is an important customer, suppliers allow the firm to stretch its credit terms.) What is the retailer's effective cost of trade credit? Assume 365 days in year for your calculations. Do not round intermediate calculations. Round your answer to two decimal places.

ANSWER: (percentage)

5. Accounts Payable

A chain of appliance stores, APP Corporation, purchases inventory with a net price of $250,000 each day. The company purchases the inventory under the credit terms of 2/15, net 40. APP always takes the discount, but takes the full 15 days to pay its bills. What is the average accounts payable for APP? Round your answer to the nearest dollar.

ANSWER: $