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linear algebra

Linear Algebra A Modern Introduction 3rd edition David Poole - Solutions

Solve the system of differential equations in the given exercise using Theorem 4.41.Exercise 63Data From Exercise 63x′ = y - z, x(0) = 1y′ = x + z, y(0) = 0z′ = x + y, z(0) = -1
Solve the system of differential equations in the given exercise using Theorem 4.41.Exercise 60Data From Exercise 60x′ = 2x - y, x(0) = 1y′ = -x + 2y, y(0) = 1
Solve the system of differential equations in the given exercise using Theorem 4.41.Exercise 59Data From Exercise 59x′ = x + 3y, x(0) = 0y′ = 2x + 2y, y(0) = 5
Use Exercise 69 to find the general solution of the given equation.x′′ + 4x′ + 3x = 0Let x = x(t) be a twice-differentiable function and consider the second order differential equationx′′ + ax′ + bx = 0
Use Exercise 69 to find the general solution of the given equation.x′′ - 5x′ + 6x = 0Let x = x(t) be a twice-differentiable function and consider the second order differential equationx′′ + ax′ + bx = 0
Show that there is a change of variables that converts the nth order differential equationx(n) + an-1x(n-1)+ ....+ a1x′ + a0 = 0into a system of n linear differential equations whose coefficient matrix is the companion matrix C( p) of the polynomial p(λ) = λn + an-1λn-1 +....+ a1λ + a0. [The
Let x = x(t) be a twice-differentiable function and consider the second order differential equationx′′ + ax′ + bx = 0(a) Show that the change of variables y = x′ and z = x allows equation (11) to be written as a system of two linear differential equations in y and z.(b) Show that the
In Exercises 67 and 68, species X preys on species Y. The sizes of the populations are represented by x = x(t) and y = y(t). The growth rate of each population is governed by the system of differential equations x€² = Ax + b, whereand b is a constant vector. Determine what happens to the
In Exercises 67 and 68, species X preys on species Y. The sizes of the populations are represented by x = x(t) and y = y(t). The growth rate of each population is governed by the system of differential equations x€² = Ax + b, whereand b is a constant vector. Determine what happens to the
Two species, X and Y, live in a symbiotic relationship. That is, neither species can survive on its own and each depends on the other for its survival. Initially, there are 15 of X and 10 of Y. If x = x(t) and y = y(t) are the sizes of the populations at time t months, the growth rates of the two
A scientist places two strains of bacteria, X and Y, in a petri dish. Initially, there are 400 of X and 500 of Y. The two bacteria compete for food and space but do not feed on each other. If x = x(t) and y = y(t) are the numbers of the strains at time t days, the growth rates of the two
Find the general solution to the given system of differential equations. Then find the specific solution that satisfies the initial conditions. (Consider all functions to be functions of t.)x′ = x + 3z, x(0) = 2y′ = x - 2y + z, y(0) = 3z′ = 3x + z, z(0) = 4
Find the general solution to the given system of differential equations. Then find the specific solution that satisfies the initial conditions. (Consider all functions to be functions of t.)y′1 = y1 - y2, y1(0) = 1y′2 = y1 + y2, y2(0) = 1
Find the general solution to the given system of differential equations. Then find the specific solution that satisfies the initial conditions. (Consider all functions to be functions of t.)x′1 = x1 + x2, x1(0) = 1x′2 = x1 - x2, x2(0) = 0
Find the general solution to the given system of differential equations. Then find the specific solution that satisfies the initial conditions. (Consider all functions to be functions of t.)x′ = 2x - y, x(0) = 1y′ = -x + 2y, y(0) = 1
Find the general solution to the given system of differential equations. Then find the specific solution that satisfies the initial conditions. (Consider all functions to be functions of t.)x′ = x + 3y, x(0) = 0y′ = 2x + 2y, y(0) = 5
In Example 4.41, find eigenvectors v1and v2corresponding to andWithverify formula (2) in Section 4.5. That is, show that, for some scalar c1, 1 + V5 2 1- V5 2.
You have a supply of 1 Ã— 2 dominoes with which to cover a 2 Ã— n rectangle. Let dnbe the number of different ways to cover the rectangle. For example, Figure 4.32 shows that d3= 3.The three ways to cover a 2 Ã— 3 rectangle with 1 Ã— 2 dominoes(a)
You have a supply of three kinds of tiles: two kinds of 1 Ã— 2 tiles and one kind of 1 Ã— 1 tile, as shown in Figure 4.30.Let tn be the number of different ways to cover a 1 Ã— n rectangle with these tiles. For example, Figure 4.31 shows that t3 =
The Fibonacci recurrence fn= fn-1+ fn-2has the associated matrix equation xn= Axn-1, whereand (a) With f0 = 0 and f1 = 1, use mathematical induction to prove thatfor all n ‰¥ 1.(b) Using part (a), prove thatfn+1fn-1 - f2n = (-1)nfor all n 1. [This is called
Complete the proof of Theorem 4.38(a) by showing that if the recurrence relation xn = axn-1 + bxn-2 has distinct eigenvalues λ1 ≠ λ2, then the solution will be of the formxn = c1λn1 + c2λn2
The recurrence relation in Exercise 45. Show that your solution agrees with the answer to Exercise 45.Data From Exercise 45y0 = 0, y1 = 1, yn = yn-1 - yn-2 for n ≥ 2
Solve the recurrence relation with the given initial conditions.b0 = 0, b1 = 1, bn = 2bn-1 + 2bn-2 for n ≥ 2
Solve the recurrence relation with the given initial conditions.a0 = 4, a1 = 1, an = an-1 - an-2/4 for n ≥ 2
Solve the recurrence relation with the given initial conditions.y1 = 1, y2 = 6, yn = 4yn-1 - 4yn-2 for n ≥ 3
Solve the recurrence relation with the given initial conditions.x0 = 0, x1 = 1, xn = 4xn-1 - 3xn-2 for n ≥ 2
Solve the recurrence relation with the given initial conditions.x0 = 0, x1 = 5, xn = 3xn-1 + 4xn-2 for n ≥ 2
Write out the first six terms of the sequence defined by the recurrence relation with the given initial conditions.y0 = 0, y1 = 1, yn = yn-1 - yn-2 for n ≥ 2
Write out the first six terms of the sequence defined by the recurrence relation with the given initial conditions.a1 = 128, an = an-1/2 for n ≥ 2
Write out the first six terms of the sequence defined by the recurrence relation with the given initial conditions.x0 = 1, xn = 2xn-1 for n ≥ 1
Explain the results of your exploration in Section 4.0 in light of and Section 4.5. The absolute value of a matrix is defined to be the matrix |A| = [ |aij| ]
(a) If A is the adjacency matrix of a graph G, show that A is irreducible if and only if G is connected. (A graph is connected if there is a path between every pair of vertices.)(b) Which of the graphs in Section 4.0 have an irreducible adjacency matrix? Which have a primitive adjacency matrix?
It can be shown that a nonnegative n Ã— n matrix is irreducible if and only if (I + A)n-1> O. Use this criterion to determine whether the matrix A is irreducible. If A is reducible, find a permutation of its rows and columns that puts A into the block form 1 0 0 0 0 0 0 0 1 A =| 1 0
Find the Perron root and the corresponding Perron eigenvector of A. 2 1 A = 1 0 1
Let L be a Leslie matrix with a unique positive eigenvalue λ1. Show that if λ is any other (real or complex) eigenvalue of L, then |λ| ≤ λ1. [Write λ = r(cosθ + i sin θ) and substitute it into the equation g(λ) = 1, as in part (b) of Exercise 23. Use De Moivre’s Theorem and then take
Find the sustainable harvest ratio for the seal in Exercise 22. (Conservationists have had to harvest seal populations when over fishing has reduced the available food supply to the point where the seals are in danger of starvation.)
(a) Find the sustainable harvest ratio for the woodland caribou in Exercise 42 in Section 3.7.(b) Using the data in Exercise 42 in Section 3.7, reduce the caribou herd according to your answer to part (a). Verify that the population returns to its original level after one time interval.Woodland
Many species of seal have suffered from commercial hunting. They have been killed for their skin, blubber, and meat. The fur trade, in particular, reduced some seal populations to the point of extinction. Today, the greatest threats to seal populations are decline of fish stocks due to over
Compute the steady state growth rate of the population with the Leslie matrix L from the given exercise. Then use Exercise 18 to help find the corresponding distribution of the age classes.Exercise 24 in Section 3.7Data From Exercise 24 Section 3.7 1/2 0 1/3 O 2/3. 1/2
Compute the steady state growth rate of the population with the Leslie matrix L from the given exercise. Then use Exercise 18 to help find the corresponding distribution of the age classes.Exercise 20 in Section 3.7Data From Exercise 20 Section 3.7 1/3 2/3] 1/2 1/2.
If all of the survival rates siare nonzero, let
Which of the stochastic matrices are regular? 0.5 1 0.5 0 1
Which of the stochastic matrices are regular? 0.1 0 0.5 0.5 1 0 _0.4 0 0.5.
Use the shifted inverse power method to approximate, for the matrix A in the given exercise, the eigenvalue closest to Exercise 13, Î± = -2Data From Exercise 13 9. 4 8 1, k = 5 15 -4 , Xo A = 4 8 -4 9.
Use the shifted inverse power method to approximate, for the matrix A in the given exercise, the eigenvalue closest to Exercise 7, Î± = 5Data From Exercise 7 4 0 6 10.000 1, X8 -1 0.001 6 0 4 10.000
Use the shifted inverse power method to approximate, for the matrix A in the given exercise, the eigenvalue closest to Exercise 12, Î± = 0Data From Exercise 12 3.5 1.5 , Xo k = 6 1.5 -0.5.
Apply the inverse power method to approximate, for the matrix A in the given exercise, the eigenvalue that is smallest in magnitude. Use the given initial vector x0, k iterations, and three-decimal-place accuracy.Exercise 7,Data From Exercise 7 1, k = 5 Xo -1 4 0 6. 10.000 1, X8 0.001 6 0 4 10.000
The power method does not converge to the dominant eigenvalue and eigenvector. Verify this, using the given initial vector x0. Compute the exact eigenvalues and eigenvectors and explain what is happening. -1 0 , Xo -1
The matrices in either are not diagonalizable or do not have a dominant eigenvalue (or both). Apply the power method anyway with the given initial vector x0, performing eight iterations in each case. Compute the exact eigenvalues and eigenvectors and explain what is happening 1, Xo 5
Use the power method to approximate the dominant eigenvalue and eigenvector of A. Use the given initial vector x0, the specified number of iterations k, and three-decimal-place accuracy. 3 1 0 1, k = 6 1 3 1 A = Xo [0 1 3, ||
Use the power method to approximate the dominant eigenvalue and eigenvector of A. Use the given initial vector x0, the specified number of iterations k, and three-decimal-place accuracy. 9. 4 8 1, k = 5 A = 15 -4 , Xo 4 8 -4
Use the power method to approximate the dominant eigenvalue and eigenvector of A. Use the given initial vector x0, the specified number of iterations k, and three-decimal-place accuracy. 3.5 |A = 1.5 Xo k = 6 1.5 -0.5. ||
A matrix A is given along with an iterate xk, produced using the power method, as in Example 4.31.(a) Approximate the dominant eigenvalue and eigenvector by computing the corresponding mkand yk. (b) Verify that you have approximated an eigenvalue and an eigenvector of A by comparing
A matrix A is given along with an iterate x5, produced as in Example 4.30.(a) Use these data to approximate a dominant eigenvector whose first component is 1 and a corresponding dominant eigenvalue. (Use three-decimal-place accuracy.)(b) Compare your approximate eigenvalue in part (a) with the
Let(a) Prove that A is diagonalizable if (a -d)2 + 4bc > 0 and is not diagonalizable if (a - d)2 + 4bc < 0.(b) Find two examples to demonstrate that if (a - b)2 + 4bc = 0, then A may or may not be diagonalizable. b. A d
Prove that if A is similar to B, then AT is similar to BT.
In general, it is difficult to show that two matrices are similar. However, if two similar matrices are diagonalizable, the task becomes easier. Show that A and B are similar by showing that they are similar to the same diagonal matrix. Then find an invertible matrix P such that P-1AP = B. -3 -2 B
In general, it is difficult to show that two matrices are similar. However, if two similar matrices are diagonalizable, the task becomes easier. Show that A and B are similar by showing that they are similar to the same diagonal matrix. Then find an invertible matrix P such that P-1AP = B. 2 -5 3 0
Prove Theorem 4.22(b).b. A is invertible if and only if B is invertible.
Prove Theorem 4.21(c).c. If A ~ B and B ~ C, then A ~ C.
Find all (real) values of k for which A is diagonalizable. k k A = k_ _1
Find all (real) values of k for which A is diagonalizable. k A = | 0 2 0
Find all (real) values of k for which A is diagonalizable. A = | 0
Use the method of Example 4.29 to compute the indicated power of the matrix. |k -2 2
Use the method of Example 4.29 to compute the indicated power of the matrix. 1k 3 1 3
Use the method of Example 4.29 to compute the indicated power of the matrix. 1 0 1
Use the method of Example 4.29 to compute the indicated power of the matrix. ]k 3
Use the method of Example 4.29 to compute the indicated power of the matrix. -5 -3 -1
Use the method of Example 4.29 to compute the indicated power of the matrix. -5 8 15 -4 7
Determine whether A is diagonalizable and, if so, find an invertible matrix P and a diagonal matrix D such that P-1AP = D. 2 0 0 0 -2 -2
Determine whether A is diagonalizable and, if so, find an invertible matrix P and a diagonal matrix D such that P-1AP = D. 1 2 1 A = -1
Determine whether A is diagonalizable and, if so, find an invertible matrix P and a diagonal matrix D such that P-1AP = D. 1 0 0 3 3 0
Determine whether A is diagonalizable and, if so, find an invertible matrix P and a diagonal matrix D such that P-1AP = D. A = 1 0]
Let (a) Find the characteristic polynomial of A.(b) Find all of the eigenvalues of A.(c) Find a basis for each of the eigenspaces of A. -5 -6 3 A = 3 4 -3 0 -2.
Determine whether A is diagonalizable and, if so, find an invertible matrix P and a diagonal matrix D such that P-1AP = D. -3 4 A -1
A diagonalization of the matrix A is given in the form P-1AP = D. List the eigenvalues of A and bases for the corresponding eigenspaces. 3 1 3 1. -1 -1 6. -2 -2 || 3. 3. 3. H|00 MI43/8
A diagonalization of the matrix A is given in the form P-1AP = D. List the eigenvalues of A and bases for the corresponding eigenspaces. 2 1 -1 3 -1 -1 2 3 3
Show that A and B are not similar matrices. -5 |A = 3 B = 4 -2 4.
Prove Theorem 4.19. Suppose the n × n matrix A has eigenvectors v1, v2, . . . , vm with corresponding eigenvalues λ1, λ2, . . . , λm. If x is a vector in Rn that can be expressed as a linear combination of these eigenvectors??say, then, for any integer k, x = qv, + Gv2 + ……+ G„Vm + GmVr
Verify the Cayley-Hamilton Theorem for A =The Cayley-Hamilton Theorem can be used to calculate powers and inverses of matrices. For example, if A is a 2 Ã— 2 matrix with characteristic polynomial cA(Î») = Î»2 + aÎ» + b, then A2 + aA + bI = O,
Prove Theorem 4.18(c). [Combine the proofs of parts (a) and (b) and see the fourth Remark following Theorem 3.9 (page 175).]a. For any positive integer n, λn is an eigenvalue of An with corresponding eigenvector x.b. If A is invertible, then 1/λ is an eigenvalue of A1 with corresponding
Prove Theorem 4.18(b).b. If A is invertible, then 1/λ is an eigenvalue of A-1 with corresponding eigenvector x.
Compute(a) The characteristic polynomial of A,(b) The eigenvalues of A,(c) A basis for each eigenspace of A,(d) The algebraic and geometric multiplicity of each eigenvalue. 1 0 0 A = 1 1 3 -2 1 -1
Compute(a) The characteristic polynomial of A,(b) The eigenvalues of A,(c) A basis for each eigenspace of A,(d) The algebraic and geometric multiplicity of each eigenvalue. 3 A = | 2 -2 [3 2.
Compute(a) The characteristic polynomial of A,(b) The eigenvalues of A,(c) A basis for each eigenspace of A,(d) The algebraic and geometric multiplicity of each eigenvalue. 4 0 A = 3 2 -1 0 2,
Compute(a) The characteristic polynomial of A,(b) The eigenvalues of A,(c) A basis for each eigenspace of A,(d) The algebraic and geometric multiplicity of each eigenvalue. -1 A = 2 -1 1.
Compute(a) The characteristic polynomial of A,(b) The eigenvalues of A,(c) A basis for each eigenspace of A,(d) The algebraic and geometric multiplicity of each eigenvalue. 1 1 A = 1 0 1 Lo 1 1
Compute(a) The characteristic polynomial of A,(b) The eigenvalues of A,(c) A basis for each eigenspace of A,(d) The algebraic and geometric multiplicity of each eigenvalue. A = | 0 -2 1
Compute(a) The characteristic polynomial of A,(b) The eigenvalues of A,(c) A basis for each eigenspace of A,(d) The algebraic and geometric multiplicity of each eigenvalue. -9 -5
A and B are n × n matrices.A square matrix A is called nilpotent if Am = O for some m > 1. (The word nilpotent comes from the Latin nil, meaning “nothing,” and potere, meaning “to have power.” A nilpotent matrix is thus one that becomes “nothing”—that is, the zero matrix—when
Prove Theorem 4.7.If A is an n × n matrix, thendet(kA) = kn det A
Prove Lemma 4.5.Let B be an n × n matrix and let E be an n × n elementary matrix. Thendet(EB) = (det E) (det B)We can use Lemma 4.5 to prove the main theorem of this section: a characterization of invertibility in terms of determinants.
Prove Theorem 4.3(f).
Find the determinants assuming that c| f = 4 |За -b 2c |за -e 2f 3g -h 2i
Use properties of determinants to evaluate the given determinant by inspection. Explain your reasoning. |1 0 1 0|
Use properties of determinants to evaluate the given determinant by inspection. Explain your reasoning. |1 0|
Use properties of determinants to evaluate the given determinant by inspection. Explain your reasoning. 4 1 3 -2 0 -2 5 4 1.
Evaluate the given determinant using elementary row and/or column operations and Theorem 4.3 to reduce the matrix to row echelon form.The determinant in Exercise 14Data From Exercise 14 -2 0 1 1 3 0 -1 2 2 3
Compute the indicated 3 Ã— 3 determinants using the method of Example 4.9.The determinant in Exercise 8Data From Exercise 8 2 3 -4 4 -3 -2 -1
Compute the determinants using cofactor expansion along any row or column that seems convenient. al |0 d e h i j

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