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Linear Algebra

Asymptotic stability of a system and Continuous Time

Determine the asymptotic stability of the system x' = Ax, where A is 2x2 matrix, A = alpha beta gamma delta ( that is. first row is alpha beta, second row is gamma delta) if it is known that determinant of A, det(A) = alpha*delta - beta*gamma > 0, and th

Eigenvalues : Asymptotic Stability of a System

Determine the asymptotic stability of the system x' = Ax where A is 3 x 3 matrix A = -1 1 1 0 0 1 0 0 -2 ( first row is -1 1 1 second is 0 0 1 and third is 0 0 -2) More specifically, what stability conclusion(s) can be drawn? ( Justify your answer)

Eigenvalues, Eigenvectors and Trajectories in the Phase Plane

(i) Find eigenvalues and eigenvectors. (ii) Classify the critical point (0, 0) as to type and determine whether it is stable or unstable. (iii) Sketch several trajectories in the phase plane. Please see the attached file for the fully formatted problems.

Systems of Differential Equations To Solve Problems

Show your argument in details you can use Maple to assist you in long calculations. YOU CANNOT USE dsolve command! Consider the following system 1) Find the general solution the systems 2) Find the solution that satisfies and . Is the solution unique? 3) Plot a (the) solution of question 2). (See attached

Exponential Random Variables with Parameters

It is known that the time (in hours) between consecutive traffic accidents can be described by the exponential r.v. X with parameter (Lambda = 1/60). Find (i) P(X < or = 60); (ii) P(X> 120); and (iii) P(10<X< or = 100).

Greatest Common Divisors

1. Let a, b be positive integers, and write a = qb + r, where q, r are Elements of Z and 0 (= or)< r < b. Suppose that d = gcd(a, b). a) If r = 0 show that d = b. b) If r > 0 show that d = gcd(b, r). 4. Use Problem 1 to find: a) gcd(100; 3); b) gcd(100; 82).

Common Divisors

1. Let d; a; b; r, and q be integers. a) Suppose that d|a and d|b. Show that d|(ra + qb). b) Suppose a = qb + r. Show that the set of common divisors of a and b is the set of common divisors of b and r.

Functions which do not converge uniformly on [0,1]

(See attached file for full problem description with proper equations) --- 9.3-3 Let . Use the result of exercise 4 of Section 9.1 to show that does not converge uniformly on [0,1], even though converges pointwise. ---

Compact Metric and Normed Spaces

Let X be a compact metric space and Y be a normed space. Prove that if f_n belongs to C(X,Y), then lim_n f_n = f_o in the Sup norm if and only if lim_n f_n = f_o uniformly in X. [ Note: Sup norm: ||f|| = Sup||f(x)|| for every x in X.]

Linear approximation

Use linear approximation, the tangent line approximation, to approximate the following: (56.4)^(1/3) (64.4)^(1/3) Show processes. Do not use a calculator. Note: the correct answer are different from the calculator computed values

Three Variable Systems of Equations from Application Problems

1. The sum of three numbers is 6. The third number is the sum of the first and second numbers. The first number is one more than the thrid number. Find the numbers. 2. Sports - Alexandria High School scored 37 points in a football game. Six points are awarded for each touchdown. After each touchdown, the team can earn

Linear algebra - Orthogonality and Projection

Q: Find the point P on the line passing through both the origin and the point 1,1,1 that is closest to the point 2,4,4. Then find the point q on the line passing through both the origin and the point 2,4,4 that is closes to the point 1,1,1

Prove the Extended Pythagorean Theorem: E^2=lbl

We've defined the error between a vector b in Euclidean vector space R^m and its projection p onto a subspace of R^m as E=lb-pl. Prove the extended pythagorean Theorem: E^2=lbl^2 - lPl^2.

Discontinuous Counter Example

I need a counterexample for the following: If f:[a,b] -> R is ONE-TO-ONE and satisfies the intermediate value property, then f is continuous on [a,b]. I know that this is a false statement if you exclude the one-to-one property. The example I received before was f(x) = sin(1/x), but this function is not one-to-one. I am

Mathematical modeling using systems of linear first order equations

Two tanks each hold 3 liters of salt water and are connected by two pipes (see figure below) the salt water in each tank is kept well stirred. Pure water flows into tank A at a rate of 5 liters per minute and the salt mixture exits tank B at the same rate. Salt water flows from tank A to tank B at the rate of 9 liters per min

Space Defined Elements

(See attached file for full problem description with symbols) --- Suppose is a matrix such that defines an element for . Show that . ---

Linear Algebra : Leontief Input-Output Model and Real-World Applications

1. Why did Leontief use linear algebra techniques to create his model? Can you think of alternative methods? 2. What are the main strengths of his model? 3. Does it have any limitations (that you can think of)? 4. How might the Input-Output model be useful in the real world? (In other words, would anyone except an Economist

Mathematical System

I am having a problem drawing the table for the following system: Define a universal set U as the set of counting numbers. Form a new set that contains all possible subsets of U. This new set of subsets together with the operation of set intersection forms a mathematical system. Then I have to tell which properties that we did

System of equations and matrices

(See attached file for full problem description with equations) --- Consider the system: a) Rewrite using matrix notation b) show that the following vectors , Are solutions and they are linearly independent c)write the general solution of the system ---