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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

### Quantitative Reasoning: Systems of Linear Equations with two unknown variables

Solve the system of equations x + y = 4 x - y = -7 Assignment Expectations: Define a system of equations. Solve systems of equations with two and three variables.

### 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

### Solving linear systems using LU factorisation

Please help me with this problem: Solve the system Ax = b by doing LU factorization. ** Please see the attached file for the complete problem description **