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# Linear Algebra

### Five Questions - Sequences, Linear equations etc.

1) Use inductive reasoning to determine the next three numbers in the pattern: 3,12,26,45.....? (I don't seem to find a pattern?) 2) Find the counterexample to show that the following statemnet is incorrect. "the sum of any odd numbers is divisible by 4. ( I tried several odd numbers and the answer is always divisible by 4)

### Seven Problems on Determinants, Cramer's Rule and Matrices

Seven practice problems are in the attached file. 1. Use Cramer's rule to solve the system x + y -z = -1 2x - 2y + 3z = -12 X + y - 4z = 13 Solution set is 2. Evaluate the determinate 5 0 0 1 -3 1 1 2 4 3. Use Cramer's rule to solve the system or to determine if it is inconsistent(I) or dependant(

### Converting temperature and modeling cost

Find a record-breaking temperature (in degrees Celsius) for a town or city in a country other than the United States. Include the name of the town and country along with the temperature, and what record was broken. Give the formula for converting degrees Celsius to degrees Fahrenheit. Using the formula, show how to convert the m

### Manipulating Basic Linear Equations

The following is offered as a solution of the equation -4[x-2(2x-3)]+1= 1/2(4x-6) -4[x-2(2x-3)]+1=8x-12 -4x-4x+6+1=8x-12 -8x+7=8x-12 7= -12 Because 7 = -12 is not a true equation, the equation has no solution. If this is correct, state that there is no solution. If not, explain in detail why it is not correct

### System of Linear Equations

(1.) x-y+2z=13 2x+xy-z=-6 -x+3y + z =-7 a. Provide a coefficient matrix corresponding to the system of linear equations. b. What is the inverse of this matrix? c. What is the transpose of this matrix? d. Find the determinant for this matrix. (2) A = [2 -3] [-4 1] [7 4] and B=[ 6 5]

### Matrices, Inverse, Transpose, Determinant, Gaussian Elimination and Cramer's Rule

Matrix methods can be used to solve linear programming problems. A linear programming problem is used to find an optimal solution, subject to stated restraints. 1. For example, consider an accountant who prepares tax returns. Suppose a form 1040EZ requires \$12 in computer resources to process and 22 minutes of the accountant'

### The Acme Class Ring Company designs and sells two types of rings: the VIP and the SST.

Attached are problems involving linear equations. Thank you for your help. 1.) The Acme Class Ring Company designs and sells two types of rings: the VIP and the SST. They can produce up to 24 rings each day using up to 60 total man-hours of labor. It takes 3 man-hours to make one VIP ring and 2 man-hours to make one SST ring

### Prove that the fields R and C are not isomorphic.

Check if the proof is correct. I need help to justify some of my answers by using Theorems, Definitions and etc. You can change my wording but try to stick to my idea. It's really important that you explain your work. Thanks! Note: R=Real Number and C=Complex Number

### Provide a plot of temperature versus distance east. Based on its appearance, is it a good linear model for predicting temperature as a function of distance?

Details: Your company was very impressed with your results from the recent study. They have asked that you investigate another location for them and present this new promising location at the upcoming sales meeting. Again, it is your job to check out the various areas and the friendliness for business. One of the factors of inte

### Word Problems Involving Linear Equations in Two Variables

1. Suppose you are in the market for a new home and are interested in a new housing community under construction in a different city. a) The sales representative informs you that there are two floor plans still available, and that there are a total of 56 houses available. Use x to represent floor plan #1 and y to represent floo

### Modeling with Linear Equations

There was several questions ask from my algebra class as a homework which I am having difficulty answering. Please help me solve for them thanks. The equation y = 1.13x + 7.85 represents the average monthly cost in dollars for cable television where x represents the number of years after 1980. Use this equation to answer the

### Solving Systems of Equations

1. y-2x=0 y=8x-9 What is the solution of the system of equations? (i need graph also) 2. r-6s=0 9r-8s=230 What is the solution of the system? 3. x+y= -13 9x+y= -61 What is the solution of the system? 4. (1,2); 6x-5y= -4 2x-7y= -12 is the given ordered pair a solution of the s

### solving system of linear equations using matrix method

1. Write the augmented matrix for the system of linear equations. a) 3x-2y+5z=31 x+3y-3z=-12 -2x-5y-3z=11 b) x-2y+3z=9 y+3z=5 z=5 3. write the system of linear equations represented by the following matrix. Use x,y, and z as variables: 4. Perform row operation and write the new matrix. 5. Solve the system of

### Word Problems and Systems of Equations

Train Tickets At the the Pittsburg zoo, children ride a train for 25 cents, adults pay \$1.00, and Senior citizens 75 cents. On a given day, 1400 passengers paid a total of \$740 for the rides. There were 250 more children riders than all other riders. Find the number of children, adult, and senior riders. Manufacturing St

### Algebra : Word Problems and Systems of Equations

Rowing Speed: Hank can row a boat 1 mi upstream (against the current) in 24 min. He can row the same distance downstream in 13min. If both the rowing speed and current speed are constant, find Hank's rowing speed and the speed of the current. Airplane Speed - An airplane flying with the wind from Los Angeles to New York Ci

### Systems of Equations and Inequalities Word Problems

Statistics. After reading an article on the front page of The New York Times titled "You Have to Be Good at Algebra to Figure Out the Best Deal for Long Distance," Rafaella De La Cruz decided to apply her skills in algebra to try to decide between two competing long-distance companies. It was difficult at first to get the compan

### Forty six questions related to finding factors, prime numbers, Greatest common factor, Least common factor, fractions, mathematical operations, solving linear equations, coordinate geometry, graphing, slope intercept form of graphs, inequality graphs, solving a system of equations and word problems.

Complete and please show all the work. Please see attached file for full problem description. 1. List all the factors of 45. 2. Which number is prime? A) 1 B) 12 C) 31 D) 99 3. List all the prime numbers between 25 and 60. 4. Find the GCF for 68, 85, and 153. 5. Find the LCM for 18 and 27. 6. Mul

### Matrix method to solve the linear system of equations

Please see the attached file. 3. A company's employees are working to create a new energy bar. They would like the two key ingredients to be peanut butter and oats, and they want to make sure they have enough carbohydrates and protein in the bar to supply the athlete. They want a total of 22 carbohydrates and 14 grams of prot

### Linear Algebra: Hyperplanes

The equation 2x_1 + 2x_2 - 3x_3 + 8x_4 = 6 defines a hyperplane in R^4. a. Give its normal vector a. b. Find its distance from the origin using dot products. c. Find the point on the hyperplane closest to the origin by using the parametric equation of the line through 0 with direction vector a. Double-check your answer in

### MTH 212: Unit 4 Group Project - A

Using the attached file for full description: 1. Along a straight shoreline, two lighthouses, A and B, are located 2000 feet apart. A buoy lies in view of both lighthouses, with angles 1, 2, and 3 as indicated. (Angle 1 is denoted by , angle 2 is denoted by , and angle 3 is denoted by .) A. By looking at the picture, d

### Linear equations and inequalities explained in this solution

Part 1: What is the formula for the volume of a rectangular solid? Find an object in your residence that has the shape of a rectangular solid. Measure and record the length, width, and height of your object in either centimeters (to the nearest 10th of a centimeter) or inches (to the nearest quarter of an inch). Compute the vol

### Equation of Motion: Spring Mass System Example

Please see attached for question and diagram. A certain engineering system can be represented by mass in a spring, as shown in Figure 1. If the mass is pulled downwards and then released, it oscillates on the spring. Using Newton's second law, a homogeneous second-order differential equation can be set up as below (see attach

### Linear Programming : Optimizing using Matrix Methods

Matrix methods can be used to solve linear programming problems. A linear programming problem is used to find an optimal solution, subject to stated restraints. For example, consider an accountant who prepares tax returns. Suppose a form 1040EZ requires \$12 in computer resources to process and 22 minutes of the accountant's t

### Matrices and linear equations

Need help in understanding what is going on with this problem: Matrices are the most common and effective way to solve systems of linear equations. However, not all systems of linear equations have unique solutions. Before spending time trying to solve a system, it is important to establish whether it in fact has a unique sol

### Linear Algebra : Orthogonal Basis

Please see the attached file for the fully formatted problems. Problem a. quote a theorem which guarantees that there exists an orthogonal basis for (with standard inner product) made up of eigenvectors of matrix b. Find such a basis . c. Represent the quadratic form by a symmetric matrix. Is Q positive definite?

### Systems of Equations and Reduced Row Echelon Form

1) Solve the system using row echelon form 2x-3y+2z=-3 Please show step by step. -3x+2y+z=1 4x+y-3z=4 1b) Solve the system y=x^2-2 & 3x-y=1

### Complementary Angles and Systems of Equations

Two angles are complementary of each other. Twice one angle is equal to the other angle plus the product of three and five. A. Set up a system of linear equations to represent the two angles. (Hint: You will need two equations and two unknowns.) B. Graph each of the equations on one rectangular coordinate system. (Hint: Y

### Linear Operators : Finite-dimensional Vector Space, Fields and Mappings

Let V be a finite-dimensional vector space. The base field F may be either R or C here. Let T, an element of the linear mapping of V to V, L(V), be an operator. Suppose that all non-zero elements of V are eigenvectors for T. Show that T is a scalar multiple of the identity map, i.e., that there is a &#955; in the Reals such

### Linear Algebra : Change of Coordinates

If you let B = {v1, v2, ..., vk} be a basis of a subspace V of ; and you let Q = (qij) be an n x n matrix such that C = {Q(v1), Q(v2,)...,Q(vk)} is a basis of V. If , what are the coordinates of v with respect to B? Also, if what are the coordinates of Q(v) with respect to C? Please see the attached file for the fully f

### Systems of Equations Word Problems

Supppose a baseball is thrown at 85 miles per hour.The ball will travel 320 ft when hit by a bat swung at 50 miles per hour and will travel 440 ft when hit by a bat swung at 80 miles per hour. Let y be the number of ft traveled by the ball when hit by a bat swung at x miles per hour.(Note: The precceding data is valid for 50 les