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# Linear Algebra

### Finding the Eigenvalue Expansion

Find the eigenvalue solution of: y^ IV = 0 y (0) = y' (0) = y (a) = y'(a) = 0

### Non-Linear ODE Systems.

1. Solve the Initial Value Problem x' = 3 x^2/3 , x(0) = 0. 2. Show that if we change the IVP to x(0) = 1, the same system , that is x' = 3 x^2/3, still has a unique solution on the interval ( minus infinity, plus infinity).

### Solving a system of 3 linear equations.

Question 1: Solve these linear equations for x, y, and z, 3x + 5y - 2x = 20 4x - 10y - z = -25 x + y - z = 5 the value of x is in the range (1) -4 <= x <= -3 (2) -2 <= x <= 0 (3) 0 <= x <= 2 (4) 2 <= x <= 4 Note: "<=" means "less than or equal to" Question 2: The value of y in Question 1 above lies in the range

### Stability and subspaces of a linear system

Please use a step-by-step method to solve the two following exercises: Exercise 1: Consider the linear system x' = Ax, where A is a 2 x 2 matrix with lamda in the diagonal as follows; A = [lamda, -2] [1 , lamda], and lamda is real. Determine if the system has a saddle, node, focus, or center at the ori

### Exponential and linear growth

During the worst periods of hyperinflation in a certain country the price of food increased at a rate of 25% per month. State whether this increase was linear or exponential if your food bill was \$150 in one month during this period what was it three months later?

### Norm on Linear System with Matrix A

We fix a norm II . II on R^n. If A is a real n x n matrix, define The norm II A II = sup { II Ax II : x E R^n, II x II =< 1}. Show that if Lamda is a real or complex eigenvalue of A, then the absolute value of lamda is less or equal to II A II.

### Linear vs. Exponential

State whether the growth is linear or exponential and solve the associated problem. The value of a house is increasing by \$1800 each year. If the house is worth \$110,000 today what will it be worth in five years?

### 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

### Different Ways to Solve Simultaneous Linear Equations

Solve the simultaneous linear equations by using the following methods a) substitution method b) Algebraic Elimination Method c) Cramer's Rule method 2x-3y= 5 3x+7y= -2 Please show all work.

### Exponential Matrix

Find the exponentials e^A for the following 2x2 matrices: 1) matrix with 1st row: [1, 2] and 2nd row: [ 0, -1 ]. 2) matrix with 1st row: [ 0, 1 ] and 2nd row: [ 1, 0 ]. 3) matrix with 1st row: [1, 4 ] and 2nd row: [ 3, 2 ].

### Find General Solution of x' = Ax

Find the general solution of the linear system x' = Ax where A is the following matrix: 1sr row: [ -2, 0, 0 ] 2nd row: [ 1, -2, 0 ] 3rd row: [0, 1, -2 ]. Please show the solution step by step.

### Proving or disproving a statement

Consider the statement that if a and b are real numbers in the open interval (0, 1), then a/[b(1 - a)] > 1. [Note: To say that a and b are in the open interval (0, 1) means that 0 < a < 1 and 0 < b < 1.] Either prove this statement (that is, prove that it's true for all real numbers a, b in the open interval (0, 1)) or giv

### 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

### How to Prove a Theorem

A simple theorem from number theory is proven to illustrate the concept of proof. Specifically, we prove that For every n>2, there exist integers x, y such that x is congruent to y mod(n) and x is not congruent to -y mod(n).

### THe Linear Equality in Linear Equations of One Variable

Could you please help me with this problem?: Scenario: You have just graduated from college, and you have started your first big project at your new job. Your boss informs you that you are responsible for the Equations and Inequalities section of the project and for presenting your ideas to the team. Prepare for the meeting b

### Difference between Scalar Multiplication & Matrix Multiplication

Can you help with the following questions: 1. What's the difference between scalar multiplication and matrix multiplication? What would be an example of the two? 2. Indicate whether the matrix is in row-reduced form. 3x + 7y - 8z = 5 x + 3z = -2 4x - 3y = 7 3. Indicate whether t

### 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

### Linear algebra/basis

Find a standard basis vector for R^3 that can be added to the set {v1, v2} to produce a basis for R^3. v1 = (-1, 2, 3) v2 = (1, -2, -2 ) Please show all work in detail.

### Linear algebra: Dimensions

Find the dimensions of each of the following vector spaces. a) The vector space of all diagonal n X n matrices b) The vector space of all symmetric n X n matrices c) The vector space of all upper triangular n X n matrices

### Linear Algebra: Spanning sets and basis

Could you clarify what constitutes a spanning set and a basis? Also how does one test to see if a set of vectors is a spanning set and if it is a basis?

### Using Substitution and Elimination to Solve the Linear Equation

Please solve the following system of equations using both of these methods (solve once using substitution and once using elimination): -4x+9y=-71; 3x-y=13 and -2x+3y=9; -4x+6y=18 Also, can you show you work so I can understand. Thank you.

### Linear Algebra: Vector Spaces

I am confused with this concept as to what constitutes a vector space. When is a set a vector space? I am familiar with the ten commandments that are required to determine a vector space but I am struggling to apply these to various sets to make the determination. Could you give me one of your concise explanations with ex

### Solve a System of Equations Without Graphing

Find the slope and the y-intercept for the graph of each equation in the given system. Use this information (and not the equations' graphs) to determine fi the system has no solution, one solution, or an infinite number of solutions. y = 7/3x + 8 9x - y = -8 The slow of the line y = 7/3x + 8 is _____. The y-interce

### Proving a Statement in Linear Algebra

Can you help me with the following: Let ubar and vbar be nonzero vectors in 2 or 3 space and let k=||u|| and m=||v||. Show that the vector Wbar=mubar +kvbar bisects the angle between ubar and vbar. Please show all steps in detail for my reference.

### Algebraic Versus Graphical Solution of Linear Equations

For solving a system of equations, is there a difference between using an algebraic method and using the graphical method? Explain why or why not, and include a numerical example to illustrate your point.

### Question

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?

### Systems of Equations and Inequalities: Example Problem

Timmy enjoys playing old-fashioned video games. He found 3 game systems on eBay and wants to purchase a Nintendo game system, an Atari game system, and a Sega arcade game system. The Sega system costs \$600 more than the sum of the Atari and the Nintendo systems. All together, the three game systems cost \$11,000. The Atari system

### Eigenvalues

** Please see the attached file for the complete problem description ** An n * n matrix A is said to be nilpotent if A^k = O for some positive integer k. Show that all the eigenvalues of a nilpotent matrix are equal to zero.

### Linear algebra problem with proof

Let R be the field of real numbers, and let D be the function on 2x2 matrices over R with values in R, such that D(AB)=D(A)D(B) for all A, B. Suppose that D([0,1;1,0])=/D([1,0;0,1]). Prove that: 1. D([0,0;0,0])=0 2. D(A) = 0 if A^2=0

### Systems of linear equations

Explain how you decide whether it is easier to solve a system by substitution or the elimination method. Response should be detailed and include an example