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Questions and Answers of Linear Algebra

Graph the following functions. Which are periodic? quasi-periodic? If periodic, what is the (minimal) period? (a) sin 4t + cos 6t (b) 1 + sin πt? (c) cos 1/2 π t + cos 1/3 π t (d) cos 1 + cos π
(a) A water molecule consists of two hydrogen atoms connected at an angle of 105o to an oxygen atom whose relative mass is 16 times that of the hydrogen atoms. If the bonds are modeled as linear unit
So far, our mass-spring chains have only been allowed to move in the vertical direction. (a) Set up the system governing the planar motions of a mass-spring chain consisting of two unit masses
Repeat Exercise 9.5.31 for fully 3-dimensional motions of the chain. In Exercise 9.5.31 So far, our mass-spring chains have only been allowed to move in the vertical direction. (a) Set up the system
Suppose M is a nonsingular matrix. Prove that λ is a generalized eigenvalue of the matrix pair K, M if and only if it is an ordinary eigenvalue of the matrix P = M-1 K. How are the eigenvectors
Suppose that u(t) is a solution to (9.77). Let N = ˆšM denote the positive definite square root of the mass matrix M, as defined in Exercise 8.4.27.(b) Explain in what sense this can serve as an
Provide the details of the proof of Theorem 9.38.
Solve the following mass-spring initial value problems. and classify as to(i) Overdamped(ii) Critically damped(iii) Underdamped, or(iv) Undamped(a)(b) (c) (d) (e) (f)
Consider the overdamped mass-spring equation If the mass starts out a distance 1 away from equilibrium, how large must the initial velocity be in order that it pass through equilibrium once?
(a) A mass weighing 16 pounds stretches a spring 6.4 feet. Assuming no friction, determine the equation of motion and the natural frequency of vibration of the mass-spring system. Use the value g =
Suppose you convert the second order equation (9.82) into its phase plane equivalent. What are the phase portraits corresponding to (a) Undamped (b) Underdamped (c) Critically damped, and (d)
(a) Prove that, for any non-constant solution to an overdamped mass-spring system, there is at most one time where u(t*) = 0. (b) Is this statement also valid in the critically damped case?
Discuss the possible behaviors of a mass moving in a frictional medium that is not attached to a spring, i.e., set k = 0 in (9.82).
(a) Determine the natural frequencies of the Newtonian system(b) What is the dimension of the space of solutions? Explain your answer. (c) Write out the general solution. (d) For which initial
Answer Exercise 9.5.5 for the systemIn Exercise 9.5.5 (a) Determine the natural frequencies of the Newtonian system (b) What is the dimension of the space of solutions? Explain your answer. (c) Write
Find the general solution to the following second order systems:(a)(b) (c) (d)
Show that a single mass that is connected to both the top and bottom supports by two springs of stiffnesses c1, c2 will vibrate in the same manner as if it were connected to only one support by a
Two masses are connected by three springs to top and bottom supports. Can you find a collection of spring constants c1, c2, c3 such that all vibrations are periodic?
Graph the following functions. Describe the fast oscillation and beat frequencies: (a) cos 8t - cos 9t (b) cos 26t - cos 24t (c) cos 10t + cos 9.5t (d) cos 5t - sin 5.2t
Justify the solution formulae (9.102) and (9.103).
Classify the following RLC circuits as (i) Underdamped (ii) Critically damped, or (iii) Overdamped (a) R = 1, L = 2, C = 4 (b) R = 4, L = 3, C = 1 (c) R = 2, L = 3, C = 3 (d) R = 4. L= 10, C = 2 (e)
Find the current in each of the unforced RLC circuits in Exercise 9.6.11 induced by the initial data
A circuit with R = l, L = 2, C = 4 includes an alternating current source F(t) = 25 cos 2t. Find the solution to the initial value problem Î¼(0) = 1.
A superconducting LC circuit has no resistance: R = 0. Discuss what happens when the circuit is wired to an alternating current source F(t) = α cos ηt.
Find the general solution to the following forced second order systems:(a)(b) (c) (d) (e) (f) (g)
(a) Find the resonant frequencies of a mass-spring chain consisting of two masses, m1 = 1 and m2 = 2 connected to top and bottom supports by identical springs with unit stiffness. (b) Write down an
Solve the following initial value problems:(a)(b) (c) (d) (e) (f)
Suppose one of the supports is removed from the mass-spring chain of Exercise 9.6.19. Does your forcing function still excite the resonance? Do the internal vibrations of the masses (i) Speed up (ii)
Find the resonant frequencies of the following structures, assuming the nodes all have unit mass. Then find a means of forcing the structure at one of the resonant frequencies, and yet not exciting
Solve the following initial value problems. In each case, graph the solution and explain what type of motion is represented.(a)(b) (c) (d)
A mass m = 25 is attached to a unit spring with k = 1, and frictional coefficient β = .01. The spring will break when it moves more than 1 unit. Ignoring the effect of the transient, what is the
Suppose the mass-spring-oil system of Exercise 9.5.38b is subject to a periodic external force 2cos2t. Discuss, in as much detail as you can, the long term motion of the mass.
(a) Does a function of the form u(t) = α cos η t - b cos ω t still exhibit beats when η ≈ ω, but α ≠ b? Use a computer to graph some particular cases and discuss what you observe. (b)
Write down the solution u(t, Î·) to the initial value problem(a) A non-resonant forcing function at frequency Î· ‰ Ï‰(b) A resonant forcing function at frequency Î· = Ï‰.(c) Show
Suppose μ(0) = 1. Find(l), μ(10), and u(20) when(a) u(k+1) = 2u(k)(b) u(k+l) = -.9u(k)(c) u(k+l) = i u(k)(d) u(k+1) = (1 - 2i) u(k)Is the system(i) stable?(ii) asymptotically stable?(iii) unstable?
A bank offers 5% interest compounded yearly. Suppose you deposit $120 in the account each year. Set up an affine iterative equation (10.5) to represent your bank balance. How much money do you have
Redo Exercise 10.1.10 in the case when the interest is compounded monthly and you deposit $ 10 each month. In Exercise 10.1.10 A bank offers 5% interest compounded yearly. Suppose you deposit $120 in
Each spring the deer in Minnesota produce offspring at a rate of roughly 1.2 times the total population, while approximately 5% of the population dies as a result of predators and natural causes. In
Find the explicit formula for the solution to the following linear iterative systems:(a) u(k+1) = u(k) - 2v(k), v(k+1) = - 2u(k) + v(k), u(0) = 1, v(0) = 0(b) u(k+1) = u(k) - 2/3 v(k), v(k+1) =
Find the explicit formula for the general solution to the linear iterative systems with the following coefficient matrices:(a)(b) (c) (d)
The Ath Lucas number is defined as(a) Explain why the Lucas numbers satisfy the Fibonacci iterative equation L(k+2) = L(k+1) + L(k) (b) Write down the first 7 Lucas numbers. (c) Prove that every
Prove that all the Fibonacci integers u(k) k > 0, can be found by just computing the first term in the Binet formula (10.17) and then rounding off to the nearest integer.
What happens to the Fibonacci integers u(k) if we go "backward in time", i.e., for A < 0? How is u(k) related to u(k)?
Use formula (10.20) to compute the kth power of the following matrices:(a)(b) (c) (d) (e)
Use your answer from Exercise 10.1.18 to solve the following iterative systems:(a)(b) (c) (d) (e)
A bank offers 3.25% interest compounded yearly. Suppose you deposit $ 100. (a) Set up a linear iterative equation to represent your bank balance. (b) How much money do you have after 10 years? (c)
(a) Given initial data u(0) = (1, l, 1)T, explain why the resulting solution u(k) to the system in Example 10.7 has all integer entries. (b) Find the coefficients c1. c2. c3 in the explicit solution
(a) Show how to convert the higher order linear iterative equationinto a first order system u(k) = Tu(t). See Example 10.6. (b) Write down initial conditions that guarantee a unique solution u(k) for
Apply the method of Exercise 10.1.21 to solve the following iterative equations: (a) u(k+2) = -u(k+1) + 2u(k), u(0) = 1, u(1) = 2 (b) 12u(k+2) = u(k+1) + u(k), u(0) = -1, u(1) = 2 (c) u(k+2) =
Starting with u(0) = 0, u(1) = 0, u(2) = 1. we define the sequence of tri-Fibonacci integers u(k) by adding the previous three to get the next one. For instance. u(3) = u(0) + u(l) + u(2) = 1. (a)
Suppose that Fibonacci's rabbits only live for eight years, [34]. (a) Write out an iterative equation to describe the rabbit population. (b) Write down the first few terms. (c) Convert your equation
Prove that the curves Ek = {Tk x | ||x|| = 1}, k = 0, 1,2,..., sketched in Figure 10.2 form a family of ellipses with the same principal axes. What are the semi-axes? Use Exercise 8.5.19.
Plot the ellipses Ek = {Tk x | ||x|| = 1} for k = I, 2, 3, 4 for the following matrices T. Then determine their principal axes, semi-axes, and areas. Use Exercise 8.5.19.(a)(b) (c)
Let T be a positive definite 2 × 2 matrix. Let En = (Tnx | ||x|| = 1], n = 0, 1,2.......be the image of the unit circle under the nth power of T. (a) Prove that En is an ellipse. True or false: (b)
Answer Exercise 10.1.28 when T is an arbitrary nonsingular 2 × 2 matrix. Use Exercise 8.5.19. In Exercise 10.1.28 Let T be a positive definite 2 × 2 matrix. Let En = (Tnx | ||x|| = 1], n = 0,
Show that the yearly balances of an account whose interest is compounded monthly satisfy a linear iterative system. How is the effective yearly interest rate determined from the original annual
Prove directly that if the coefficient matrix of a linear iterative system is real, both the real and imaginary parts of a complex solution are real solutions.
Explain why the solution u(k), k > 0, to the initial value problem (10.6) exists and is uniquely defined. Does this hold if we allow negative k < 0?
Prove that if T is a symmetric matrix, then the coefficients in (10.9) are given by the formula Cj = aTVj/vTjvj.
Explain why the jth column cj(k) of the matrix power Tk satisfies the linear iterative system cj(k+1) = Tcj(k) with initial data cj(0) = ej, the jth standard basis vector.
Let z(k+1) = λz(k) be a complex scalar iterative equation with λ = u + iv. Show that its real and imaginary parts x(k) = Rez(k), y(k) = Imz(k) satisfy a two-dimensional real linear iterative
Let T be an incomplete matrix, and suppose w1,... , wj is a Jordan chain associated with an incomplete eigenvalue Î».(a) Prove that, for any i = 1,... , j,(b) Explain how to use a Jordan
Use the method Exercise 10.1.36 to find the general real solution to the following iterative systems: (a) u(k+1) = 2u(k) + 3v(k), v(k+1) = 2v(k) (b) u(k+1) = u(k) + v(k), v(k+1) = -4u(k) + 5v(k) (c)
Find a formula for the fcth power of a Jordan block matrix. Use Exercise 10.1.36.
Suppose are two solutions to the same iterative system u(k+l) = T u(k).(a) Suppose for some k0 > 0. Can you conclude that these are the same solution: for all kn.(b) What can you say if ?
Show that, as the time interval of compounding goes to zero, the bank balance after k years approaches an exponential function erk a, where r is the yearly interest rate and a the initial balance.
An affine iterative system has the form u(k+1) = T u(k) + b. u(0) = c.(a) Under what conditions does the system have an equilibrium solution u(k) = u*?(b) In such cases, find a formula for the
A well-known method of generating a sequence of "pseudo-random" integers x0, x1. . . . . . . in the interval from 0 to n is based on the Fibonacci equation u(k+2) = u(k+1) + u(k) mod n. with initial
Let u(t) denote the solution to the linear ordinary differential equation Let h > 0 be fixed. Show that the sample values u(k) = u(k h) satisfy a linear iterative system. What is the coefficient
For which values of A does the scalar iterative system (10.2) have a periodic solution, meaning that u(k+m) = u(k) for some ml
Investigate the solutions of the linear iterative equation u(k+1) = λu(k) when A is a complex number with |λ| = 1, and look for patterns.
Consider the iterative systems u(k+1) = λu(k) and v(k+1) = u v(k) where |λ| > |u|. Prove that, for any nonzero initial data u(0) = a ≠ 0. v(0) = b ≠ 0, the solution to the first is eventually
Let λ, c be fixed. Solve the affine (or inhomogeneous linear) iterative equation u(k+l) = λu(k) + c, u(0) = a. (10.5) Discuss the possible behaviors of the solutions. Write the solution in the
Determine the spectral radius of the following matrices:(a)(b) (c) (d)
Discuss the asymptotic behavior of solutions to an iterative system that has two eigenvalues of largest modulus, e.g., λ1 = - λ2, or λ1, = λ2 are complex conjugate eigenvalues. How can you detect
Suppose T has spectral radius p(T). Can you predict the spectral radius of c T + dI. where c, d are scalars? If not, what additional information do you need?
Let A have singular values σ1, > ∙ ∙ ∙ > σn. Prove that ATA is a convergent matrix if and only if σ1 < 1. (Later we will show that this implies that A itself is convergent.)
Let Mn be the n × n tridiagonal matrix with all l's on the sub- and super-diagonals, and zeros on the main diagonal. (a) What is the spectral radius of Mn? Use Exercise 8.2.47. (b) Is Mn
Let α, β be scalars. Let Tα,β be the n × n tridiagonal matrix that has all or's on the sub- and superdiagonals, and β's on the main diagonal. (a) Solve the iterative system u(k+l) =
(a) Prove that if |det T| > 1 then the iterative system u(k+l) = T u(k) is unstable. (b) If |det T| < 1 is the system necessarily asymptotically stable? Prove or give a counterexample.
True or false: (a) p(cA) = cp(A) (b) p(S-1AS) = p(A) (c) p(A2) = p(A)2 (d) p(A-1) = l/p(A) (e) p(A + B) =p(A) + p(B) (f) p(A B) = p(A) p(B)
True or false: (a) If A is convergent, then A2 is convergent. (b) If A is convergent, then AT A is convergent.
Suppose Tk → A as k → ∞. (a) Prove that A2 = A. (b) Can you characterize all such matrices A? (c) What are the conditions on the matrix T for this to happen?
Determine whether or not the following matrices are convergent:(a)(b) (c) (d)
Prove that a matrix with all integer entries is convergent if and only if it is nilpotent, i.e., Ak = O for some k. Give a nonzero example of such a matrix.
Consider a second order iterative scheme u(k+2) = Au(k+l) + Bu(k). Define a quadratic eigenvalue to be a complex number that satisfies det(λ2I - λA - B) = 0. Prove that the system is asymptotically
Let p(t) be a polynomial. Assume 0 < λ < u. Prove that there is a positive constant C such that p(n) λn < C un for all n > 0.
Prove the inequality (10.28) when T is incomplete. Use it to complete the proof of Theorem 10.14 in the incomplete case. Use Exercises 10.1.36, 10.2.22.
Suppose that M is a nonsingular matrix. (a) Prove that the implicit iterative scheme M u(n+l) = u(n) is asymptotically stable if and only if all the eigenvalues of M are strictly greater than one in
Find all fixed points for the linear iterative systems with the following coefficient matrices:(a)(b) (c) (d)
Suppose T is a symmetric matrix that satisfies the hypotheses of Theorem 10.17 with a simple eigenvalue Î»1 = 1. Prove the solution to the linear iterative system has limiting value
Which of the listed coefficient matrices defines a linear iterative system with asymptotically stable zero solution?(a)(b) (c) (d) (e) (f) (g)
(a) Under what conditions does the linear iterative system u(k+1) = T u(k) have a period solution, i.e.. u(k-2) = u(k) ≠ u(k+1)? Give an example of such a system. (b) Under what conditions is there
Prove Theorem 10.18(a) Assuming T is complete,(b) For general T. Use Exercise 10.1.36.
(a) Determine the eigenvalues and spectral radius of the matrix(b) Use this information to find the eigenvalues and spectral radius of
(a) Show that the spectral radius ofis p(T) = 1. (b) Show that most iterates u(k) = Tku(0) become unbounded as k †’ ˆž. (c) Discuss why the inequality ||u(k)|| (d) Can you prove
Given a linear iterative system with non-convergent matrix, which solutions, if any, will converge to 0?
Suppose T is a complete matrix. (a) Prove that every solution to the corresponding linear iterative system is bounded if and only if p(T) < 1. (b) Can you generalize this result to incomplete
Suppose a convergent iterative system has a single dominant real eigenvalue λ1. Discuss how the asymptotic behavior of the real solutions depends on the sign of λ1.
Compute the ˆž matrix norm of the following matrices. Which are guaranteed to be convergent?(a)(b)(c)(d)(e)(f)(g)(h)

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