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# Linear Algebra

### find eigenvalues and vectors

See the attached file, please. Given the following matrices A, B, and C compute the eigenvalue and eigenvector for each matrix. A =(-2, 6, 6, 3) B= (4, 2, -1, 6) C= (0, -1, 1, 2)

### Linear equations and word problem

Please help with the following problem. 1. Sharing fruit Apples are collected in a basket for six people. One third, one fourth, one eighth, and one fifth of the apples are given to four people, respectively. The fifth person gets ten apples, and one apple remains for the sixth person. Find the original number of apples in

### Analyze linear equations.

Determine whether the lines will be perpendicular when graphed. 3x - 2y = 6 2x + 3y = 6 Solve the system of equations by the substitution method x + 3y = 32 -3x + 2y = 3 Solve the system of equations x + y = 5 x - y = -9

### Systems of Equations & Graphing

Please help solve the problems below. Make sure you show all your work so I can learn how to get the answer in the future. 1. Best Rentals charges a daily fee plus a mileage fee for renting its cars. Barney was charged \$123.00 for 3 days and 300 miles, while Mary was charged \$216.00 for 5 days and 600 miles. What does Best

### Linear Algebra: Vector Space Problems

Please see attachment. The set of elements belonging to R^2 is usually denoted as {(a, b) | a, b ∈ R}. Combining elements within this set under the operations of addition and scalar multiplication should use the following notation:   Addition Example: (-2, 10) + (-5, 0) = (-2 - 5, 10 + 0) = (-7, 10)

### Characteristic Polynomials

Please show all work. In this exercise find the following:(the characteristic polynomial of A, and the eigenvalues of A, and a basis for each eigenspace of A, the algebraic and geometric multiplicity of each eigenvalue. 1) [1 -9 1 -5] 2) [1 1 0 1 0 1 0 1 1 ] 3) [ 1 1 -1 0 2 0 -1 1 1] 4) [ 1 0 3 2 -2 2 3

### Every Number of the Family of Functions

r(n) means "r subscript n" The function y=e^(rx) has two values, r(1) and r(2) that satisfy the equation 2y''+y'-y=0. How can I show that every number of the family of functions y=a*e^[r(1)x] + b*e^[r(2)x] is also a solution to the equation 2y''+y'-y=0?

### Equations are solved.

1. Solve for X and Y in the following problems using either substitution or elimination methods. Make sure you show all your work so you can get partial credit even if your final answer is wrong. a. X + Y = 10 , 3X + Y = 12 b. 2X + 5Y = 19 , 3X + 3Y = 15 c. 4X + Y = 22 , 2X + 3Y = 16 d. 12X + Y = 174 , 8

### Equation of a Line that is Perpendicular

consider the line -8x-2y=-5 what is the slope of a line perpendicular to this line and what is the slope of the line parallel Suppose that the weight in pounds of an airline is a linear function of amount of fuel (in gallons) in its tank when carrying 20 gallons of fuel, the airplane weighs 2030 pounds, when carrying 54 gall

### Algebra/World Problems/Graphs

1) 4m+12=6m-2 Solve for m 2) 4x-7y=28 Solve for y 3) A jazz concert brought in \$179,500 on the sale of 8,700 tickets. If the tickets were sold for \$15 and \$25each, how many of each type of ticket were sold? 4) Graph the equation x+2y= 4 5) Find the slope of the graph of the equation 7x+3y= 21 6) Write the equatio

### Counting, Functions and Sets

1. Evaluate the function f(x) = 4x + 6 for x=4. 2. Evaluate the function f(x) = 9x - 6 for x=0. 3. Take a look at the following table: x -2 -1 0 1 2 f(x) -5 -2 1 4 7 a. Write out an equation for f(x). Assume the function is linear b. What is the slope? Is it negative or p

### Evaluate as linear or exponential growth.

These are the sales data for two companies A and B over the years. Year 2000 2001 2002 2003 2004 A \$110 \$120 \$130 \$140 \$150 B \$70 \$100 \$130 \$160 \$190 For each company A and B determine whether it is witnessing a linear or exponential growth in sales and explain how you arriv

### Find the slope of the line shown and interpret the slope.

Find the slope of the line shown and interpret the slope as a rate of change given the following information. I am supposed to present this in business terms using Exxon Mobile as the example and using the mathematical concept of slope and rate of change. After submitting this question earlier I am not sure the answer is what I

### product of a linear and cubic factor

(i) Determine the remainder when 9x^5 â?" 4x^4 is divided by 3x â?" 1. (ii) Show, using the factor theorem, that 2x â?" 1 is a factor of 2x^4 â?" x^3 â?" 6x^2 + 5x â?" 1 and hence express 2x^4 â?" x^3 â?" 6x^2 + 5x â?" 1 as a product of a linear and cubic factor. SEE ATTACHED FILE:

### Truth tables are exemplified.

1. Let p, q, and r be the following statements: p: Sylvia is at the park. q: Jamie is on the train. r: Nigel is in the car. Translate the following statement into English: (~q / p) -> ~r 2. Construct a truth table for ~q -> p. 3. Construct a truth table for (p / ~q) &#8596; q. 4. Let p

### Boolean product of matrices, One to one function

** Please see the attachment for complete questions (1 and 2) ** 1. Find the Boolean Product of matrices A and B. 2. Given matrix A, compute: (a) A^-1 (A-inverse) (b) (A^-1)^3 3. Solve the following systems of equations. x1 + x2 = 0 -x1 + x2 + x3 = -1 -1x2 + x3 = 2 4. (a) Define the function f: R -->

### Show divisibility.

1. Emperor's Banquet You have been invited to the emperor's banquet. The emperor is a rather strange host. Instead of sitting with his guests at a large round dining table, he walks around the table pouring oatmeal on the head of every other person. He continues this process, pouring oats on the head of everyone who has n

### System of Equations, Index Numbers

Solve the system of equations: 3x + y = -7 x - y = -5 Solve the system of equations: 3x + 2y = 18 y = 3x Which of the following is true of a base period for an index number? The numerator spears It appears in the denominator It must have occurred after the year 1980 It cannot be less than 100

### Algebra: vector space

Determine which of the following subsets of M22 form subspaces. a) The subset having diagonal elements nonzero. b) The subset of matrices whose (1,2) element is 0. c)The subset of matrices of the form ( a b ) (-b c ) d)The subset of 2x2 matrices whose determinants

### Linear algebra - vector space.

Let V be the set of functions having the set of real numbers as domain whose graphs pass through the point : a) (2,0) b) (2,1) Is V a Vector space?

### Solution sets in Algebra are assessed.

How many solution sets do systems of linear inequalities have? Must solutions to systems of linear inequalities satisfy both inequalities? In what case might they not? Use an example for discussion.

### Is a weak contraction a contraction?

Let M be a subset of a metric space (X,d). A function f: M -----> M is a weak contraction provided that for any x not equal to y in M, d ( f(x), f (y) ) < d ( x, y). a) Is a weak contraction a contraction? ( Proof or counterexample) b) If M is compact. Is a weak contraction a contraction? ( Proof or counterexample)

### preserve distance for translation map

In R2, let C be any point. Let T_c : R2 --> R2 be the translation map T_c(P) = P + C. (a.) Show T_c is a bijection by showing it is one to one and onto. (b.) Show T_c is a bijection by finding its inverse. (c.) Show T_c is distance preserving (using vectors).

### Proving that a closed unit is not compact

Prove that the closed unit ball of the normed space (C[0,1], | |infinity) is not compact. As usual C[0,1] stands for the space of all continuous functions f: [0,1] -> R, and | |infinity is the uniform norm on that space. Please refer to the attachment for appropriate notations in the question.

### Proof for Non-Empty Subsets

Please help with the following problem. Use the following steps to prove that every non-empty open subset of R is a union of at most countably many disjoint open intervals. Suppose that G is a non-empty open subset of R. 1. For each a <- G let Ia be the union of all those open intervals I which contain a and are containe

### Accurately Sketching a Phase Portrait

Classify the fixed point at the origin and accurately sketch a phase portrait for each of the following systems (i) dx/dt=x-2y dy/dt=3x-4y (ii) dx/dt=3x+2y dy/dt=-2x-2y (iii) dx/dt=-x-5y dy/dt=x+3y (iv) dx/dt=x+y dy/dt=-2x-y (v) dx/dt=2x dy/dt=-

### Assessing the Field of Quotients

Let R = Z[x]. Let f : R -----> C be defined by evaluation of polynomials at sqrt (-5), which we know to be a homomorphism. a. What is the kernel of f? why? b. the image of f is a subring of C. describe it and its field quotients. No proof.

### Irreducibility

a. Show that 1 + i is not a unit in Z[i] b. Show that 2 is not irreducible in Z[i] c. Show that 3 is irreducible in Z[i]

### Create three sets, set A, set B, and set C by going through the items in your wallet or purse.

Create three sets, set A, set B, and set C by going through the items in your wallet or purse. Set A will be a list of all of these items. Create Set B, from the items in Set A that you think are essential. Create Set C, by taking the complement of Set B in Set A, i.e. all of the non-essential items in your wallet or

### Find all the eigenvalues of a matrix

Please see attachment for full problem description and hint. The solution of the vector-valued differential equation dX/dt = AX(t) with X(0) = [a1 a2 a3] is given by: X(t) = Exp(tA)X(0) = e^tA(X(0)) Explain how you would compute e^