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Linear Algebra

Polynomials - Horner's Method, etc.

a) Given a polynomial P(x) and a point xo, what 2 things does Horner's method give us? How is the result useful for polynomial root-finding P(x)=0? b) One use of polynomials is interpolation of given data points {(xk, fk)} , k=0,....,n 1. Write down the Lagrange building block Ln,k (x)- that is the nth degree polynomial wh

Fundamental Mathematics: Rotation & Reflection in three dimensional space

We use matrices, eigenvalues and eigenvectors to solve the following: Let f be the rotation in the 3-dimensional space about the x-axis through the angle pi/2 and g be the rotation about the y-axis through the same angle. Describe the rotation f g. Let h be the reflection in the plane orthogonal to the vector 2i - j + 3k. D

How to find generalized eigenvectors

I have difficulties understanding an easy method to find the generalized eigenvectors of a nilpotent matrix. For example, for the matrix A= 1st : 1 0 0 2nd; -1 2 0 3rd: 1 1 2 The first eigenvalue is 1, and the

Condition for a Linear Subspace E of R^n to be A-Invariant

The exercise is as follows: a) Let E be a linear subspace of R^n. Show that E is a closed subspace of R^n. b) Let A be a real n x n matrix. Show that a linear subspace E of R^n is A-invariant if and only if E is e^{tA} -invariant for all t in R, where e^{tA} is the exponential matrix associated to A.

Numerical analysis and Gaussian quadrature

A function of two variables f(x,y) is integrated over the square [0,2] x [-1,1]. ex: integral from 0 to 2 , integral from -1 to 1 f(x,y) dx dy. (I wanted to input integral symbols there but didn't know how). Build a 25- point numerical integration scheme based on 5- point Gaussian quadrature in the x and y directions. Specify

Systems and Nutrition

Gather nutritional information for two different foods to devise a meal plan that provides optimal nutritional value. To maintain health, people need to consume a recommended daily amount of many different vitamins and minerals, as established by government health agencies. To achieve those levels, people have their choice of

Exploitation Model System.

** Please see the attached file for the complete problem description ** Please help me with these questions. Given an exploitation model system: a) nondimensionalize the system b) find an equilibrium point c) perform a linear perturbation analysis of the equilibrium point. d) represent the loci graphically e) sketch a

Making linear equations out of word problems.

1. Assume that you do not know how many people from your contact list will call you this week. Let's call this number z. a. What would this z be called in mathematics? A variable b. If you know that on the week of your birthday you will get 2.3 times the number of calls as normal, write an expression that writes this in terms

Linear Algebra - Formal Power Series

Question 1: Let x be a variable. Define a formal power series in the variable x over a field F to be a sum of the form a_0 + (a_1)(x) + (a_2)(x^2) + (a_3)(x^3) + ... = SUM (t=0, infinity) (a_t)(x^t) with a_t is a real number of F. Let F[[x]] be the set of all formal powers series in x over F. Define an addition on F[[x]] b


Without graphing, can you use the slope to determine whether there is a solution for a system of two linear equations in two unknowns? How?

Minimal linearly dependent sets of columns

Given two matrices {1 2 3 3 9 1 0 1 0 2 0 2 2 3 7 2 4 6 6 18} {9 3 9 3 9 1 3 3 9 0 2 5 6 0 0 2 3 1 3 1} for each one: a) list all minimal linearly dependent sets of columns; b) list all maximal linearly independent sets of columns; c) list all minimal sets of columns which span all columns; a',b',c') The same

Math 133 Unit 1 Individual Project

You will need to prepare your answers in MS Word using the student answer form provided. Click here for the answer form. Show all work for full credit; in many cases, you will be able to type your answer without special characters, fonts, or equations, but in a few cases, these features of MS Word will make your work more pres

Help with Algebra Problems

1. Add. (-9 + 3n6 + 3n5) + (2n6 + 5n5 + 6) A) 5n6 + 8n5 - 3 B) 10n11 C) 5 + 8n6 - 3n5 D) -7n6 + 8n5 + 9 2. Subtract. (8n7 + 2n6 + 17) - (5n6 + 5n7 + 15) A) 3n7 - 3n6 + 32 B) 3n7 + 7n6 + 32 C) 2n13 D) 3n7 - 3n6 + 2 3. Multiply. 4(5x) A) 20 B) 20x C) 9x D) 9 4. Multiply. -8x2(-10x4 + 9x3) A)

Linear Equations

1. Bill has $2,000 in his savings account and x dollars in his checking. Write an expression for the total amount of money Bill has. 2. You want to order T-shirts for your organization. The price is $15 for each T-shirt and $10 for shipping. Write an expression for the amount of money this will cost you. If you want to buy 20

Solve a system of linear equations with 3 variables

Cost of CDs : The accompanying table shows the total cost of purchasing combinations of differently priced CDs. The types of CDs are labeled A, B, and C. A B C Total Cost 1 1 1 $37 3 2 1 $69 1 1 4 $82 (a) Let x be the cost of a CD of type A, y be the cost of a CD

Linear Algebra, Vector Space and Mapping

See the attached file. Let Beta = {x_1, ..., x_n} be a basis for a vector space V, and let P be the mapping P((a_1)(x_1) + ... + (a_n)(x_n)) = (a_1)(x_1) + ... + (a_k)(x_k). a) Show that Ker(P) = Span({x_k+1, ..., x_n}) and Im(P) = Span ({x_1, ..., x_k}) b) Show that P^2 = P c) Show conversely that if P:V --> V is an

Question about Inequalities and Equations

1.) 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? Provide an example and a reference. 2.) Do the equations x = 4y + 1 and x = 4y - 1 have the same solution? How might you explain your answer to someone w


The techniques for solving linear equations and linear inequalities are similar, yet different. Explain and give an example of both a linear equation and a linear inequality that demonstrates this difference. 1.) Solve and check the linear equation. 5x - 5 = 30 A) {30} B) {34} C) {11} D) {7} 2.) Solve and check th

Algebra Problems & Solving Linear Equations

When solving a linear equation in one variable, the objective is to isolate the variable on one side of the equation. What does that mean? Give an example. 1. Add. (9a3 + 3a2) + (5a3 + 6a2) A) 14a6 + 9a4 B) 23a5 C) 14a3 + 9a2 D) 23a10 2. Add. (-9 + 3n6 + 3n5) + (2n6 + 5n5 + 6) A) 5n6 + 8n5 - 3 B) 10n11 C)

Rational and Irrational Number Proof

Prove or disprove each of the following: a. The sum of a rational number and an irrational number is an irrational number. b. The product of two rational numbers is a rational number. c. The product of two irrational numbers is an irrational number. d. The product of a rational number and an irrational number is an irra

Matrices Eigenvalues, eigenspace, and eigenbasis.

Matrices Eigenvalues, eigenspace, and eigenbasis. See attached file. For each of the matrices, find all (real) eigenvalues. Then find a basis of each eigenspace, and find an eigenbasis, if you can. [■(7&8@0&9)] [■(1&1@1&1)] [■(6&3@2&7)] [■(0&-1@1&2)] [■(1&1&0@0&2&2@0&0&3)] [■(1&1&0@0&1&1@0&0&1)]

Manufacturer produces three types of radios

First equation: A company makes three products X, Y and Z. Each product requires processing by three machines A, B and C. The time required to produce one unit of each product is shown below. PRODUCT MACHINE A MACHINE B MACHINE C X 1 2 2 Y 2 8 3 Z 2 1 4 The machines are available for 200, 525 and 350

Linear Spaces and Isomorphism

See the attached file. 1. Find a basis B of R^n such that the B-matrix B of the given linear transformation T is diagonal. Reflection T about the line in R^n spanned by [2;3 2. Which of the subsets of P_2 given below are subspaces of P_2? Find the basis for those that are subspaces. i) {p(t):p(0)=2} ii) {p(t):p(2)=0} i

Systems of Equations Homework Help

Solve the following systems of equations using the method outlined in week 7 (Part I) of the notes. Your procedure should be in matrix form as was done in Example 3. (Read your notes carefully before attempting the problem and simply follow the examples) x1 + x2 = 2 -x1 + x2 + x3 = 0 -1x2 + x3 = 1

Householder Reflector - Proof in Numerical Linear Algebra

1. Determine the eigenvalues, determinant, and singular values of a Householder reflector. Give algebraic proofs for your conclusions. 2. Suppose Q E C^n, llqll2 = 1 Set P = I - qq^H. (a) Find R(P) (b) Firrd l/(P). (c) Find the eigenvalues of P. Prove your clairns. 3. Let A E C^(m*n)}, m (greater or equal to) n, with ran

Problems Involving Linear Equations and Functions

See the attached file. Race was 46.2 sec in 1920. In 1940 was 46.0 sec. Let R (t) = the record in the race and t= the number of years since 1920. I need help finding R (t) = what by rounding to nearest hundredth Predicted record for 2003 is ____ sec (round to nearest hundredth) Predicted record for 2006______sec (rou

Systems of Linear Equations

Module 3 - Case Systems of Linear Equations Answer the question and solve the problems below. Make sure you show all your work so you can get partial credit even if you get the final answer wrong. 1. Determine whether the lines will be perpendicular when graphed. 3x - 2y = 6 2x + 3y = 6