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Linear Algebra

Derivatives and Systems of Linear Equations

Derivatives Use the definitions of the derivative to find the derivative of each of the given functions. y = x^3 + 5 Find the slope of the tangent line to the given curve at the given value of x. Find the equation of each tangent line. y = 8 - x^2; at x = 1 Find the derivative of the given function. G(t) = (t^3+ t-2)

Linear Programming: Blending of Fertilizers

A landscaper blends his own fertilizer which consists of four compounds at the listed price per lb: COMPOUND Cost per lb C30 $0.12 C92 $0.09 D21 $0.11 E11 $0.04 Formulate an LP problem to determine what blend of thes

Writing Linear Equations

1. Find the equation of the line. Write the equation using function notation. Please show work. Through (4,-5); perpendicular to 6y=x-12 The equation of the line is f(x) =____________________. 2. Please Show all work: In 2006, the median price of an existing home in some country was approximately $230,450. In 2001, th

ANOVA, Correlation Coefficient, and Linear Equation

See the attached file. Probability & Statistics: Resolve the following problems of regression and ANOVA. Please use the statistics program to resolve the problems below. Please explain your answers. Problem 1: Six samples of four types of milk were analyzed to determine protein content. The results where the following (ug/

Using Systems of Linear Equations to Solve Word Problems

Hello, I need help understanding how to apply linear systems to the following word problems: 1. The M-Disco company learns that each van now costs 13,000 and each small truck $18,000. The company decides to buy only 182 new Vehicles. How many of each kind should it buy? 2. Matt took his clothes to the cleaners three times

Matrix Form: Inhomogeneous Differential Equations

How do I express the following inhomogeneous system of first-order differential equations for x(t) and y(t) in matrix form? (see the attachment for the full question) x = -2x - y + 12t + 12, y = 2x - 5y - 5 How do I express the corresponding homogeneous system of differential equations, also in matrix form? How do I fin

Linear Algebra: Interest Rates and Cramer's Rule

For each output level Y, the IS curve defines the interest rate r at which the goods market clears: Y(1-b)-G=I^0-ar, where b is the marginal propensity to consume, G is the government spending, I^0 is the maximum investment level, and a is the responsiveness of investment to interest rates. The LM curve defines the intere

Demand and Production, Zero-Profit, Output Prices

Please see the attached file for complete equation. Consider an open Leontief system with three fixed-proportions production sectors. Assume that the vector of labour requirements is a^T_0 =(1,1,1), i.e. 1 unit of labour is needed to produce 1 unit of each output. The input-output matrix is A=[ 0.3, 0, 0.4 0.6, 0.1,

Linear Algebra Question: Matrices and Symmetry

Please see the attached file for full details. Assume that A and B are two symmetric k x k matrices. a) Show that AB = (BA)^T b) Show that if A is invertible, then A^-1 is also a symmetric matrix. c) Assume that we can write A= [ A_1 A_2 ] A_3 A_4 With A_1 a 2 x 2, A_2 an m x 3, A_3 a 3 x n and A_4 a

Addition Method, Linear and Compound Inequality, Absolute Value

Set 5 #22 The addition method. Solve each by addition 2x = 2 - y 3x + y = -1 #26 Solve each by addition method. Determine whether the equations are independent, dependent or inconsistent X - y = 3 -6x + 6y = 17 #30 -3x + 2y = 8 3x + 2y = 8 #36 Equations involving fractions or decimals. Solve each syste

Algebra Functions - Linear and Quadratic

Show all work in a Word document for full credit. 1. Given that f(x)= {█(-x+4,@4-x^2,@2x-6,)┤ ■(for x≤0,@for 0<x≤2,@for x>2.) Find f(-1), f(0), f(2), f(3). In problems 2 through 4, determine whether the function is even, odd, or neither. Explain the process used to make that determination. 2. f(x)=-3x

Fundamental Mathmatics

1. a) State the Lagrange Theorem explaining any terms you use. b) Let alpha, a member of S_11, be the permutation given by alpha(1) = 7, alpha(2) = 5, alpha(3) = 1, alpha(4) = 2, alpha(5) = 8, alpha(6) = 9, alpha(7) = 10, alpha(8) = 4, alpha(9) = 11, alpha(10) = 3, alpha(11) = 6. Decompose the permutation alpha first

Mixed Linear Algebra Problems

Linear equations: A) 3x - 5(6 - 2x) = 4(x - 8) + 3 B) (w - 3)/8 - (5 - w)/4 = (4w - 1)/8 - 1 C) a = 1 /2(b1 + b2) For each formula express y as a function of x: D) y -3 = 1 / 3(x-4) Solve for the specified variable: E) xy + 5 = x + 7 for x Solve the equation. Round answers to three decimal plac

Compute the Eigenvalues of the Matrix

Consider the matrix A = 3 0 0 1 5 1 -2 -4 1 (a) Show that 3 is an eigenvalue of A with algebraic multiplicity 3. (b) Determine the geometric multiplicity of this eigenvalue. (c) Find a basis for the eigenspace E3 = (A - 3I). (d) Find bas

Possible eigenvalues for a matrix A satisfying A^2=-A

Let A be an nxn matrix such that A^2=-A. Then the possible values of the eigenvalues of A are i) 0, ii) 1, iii)-1, iv) i, v) -i A. i) and ii) only B. i) and iii) only C. iv) and v) only D. i), iii), iv), and v) E. All of these

Symmetric Matrix with Positive Eigenvalues

Suppose that both eigenvalues of a 2 x 2 symmetric matrix B are positive. Which of the following statements are true? i) The system Bx = b has a unique solution for each b. ii) B is positive definite. iii) The system dx = Bx is stable. __ dt A. i) only B. ii) only C. iii)

Determining Linearly Independent Polynomials

Which of the following sets of polynomials are linearly independent? i) x-1, x-2, x-3 ii) x-1, x^(2)-2x, x^(2)-x-1 iii) 1, x+1, x^(2)+x+1 Options: A) i only B) ii only C) iii only D) i and ii E) ii and iii

Simplex Method and Gauss-Jordan elimination method.

1) Using the simplex method, solve the following linear programming equation: Maximize: P= 5x + 2y Subject to: 4x + 3y <= 30 2x ? 3y <= 6 x>=0 , y>=0 2) Solve the system of linear equation using the Gaus- Jordan elimination method. a) 3x ? 3y + 4z

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

Finding the general solution of a linear system

Please solve the following ODE problem and show the step by step method: Let the matrix A be: A = 1st row: 1 0 0 2nd row: 0 2 1 3rd row 0 0 2 Find the general solution of the linear system x' = Ax.

Hyperbolic and non-hyperbolic equilibrium points

Please solve the following problem; consider the nonlinear system x' = f(x), where f(x) = 1st row: x2 - x1 2nd row: kx1 - x2 - x1 x3 3rd row: x1 x2 - x3 a) For what values of k is the origin a hyperbolic equilibrium point of the system ? In this case, classify it as a sin

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

Fundamental Mathematics: Eigenvectors and Matrices

1) Find the eigenvectors and eigenvalues of the matrix A. Hence find the matrix P and a diagonal matrix D such that A = P^?1 DP and compute A^100 (see attached). 2.i) Verify the Cayley-Hamilton theorem for the matrix A. ii) Compute the minimal polynomial of a matrix A and decide whether the matrix is diagonalizable or not.

Properties of system du/dt=Au

Let [equation attached]. Circle the letters corresponding to the statements that are true for the system du/dt = Au. A. The system is unstable when a > 0. B. The system is stable when a < O. C. The system is neutrally stable when a = -2. D. The system is neutrally stable when a = 2. E. Trajectories of the system oscillate

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