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mathematics
linear algebra and its applications

Questions and Answers of Linear Algebra And Its Applications

The number of innings batted in the Major Leagues in the 2006 season was 43,257, and the number of earned runs scored was 21,722. What is the total number of earned runs scored for the season
Consider a pair of Ehrenfest urns labeled A and B. There are currently no molecules in urn A and 5 in urn B. What is the probability that the exact same situation will apply after a. 4
The sums of the first columns of M for the player data in Table 6 and the first columns of SM for the player data in Table 6 given in Table 7. Find and compare the offensive earned run averages of
Consider a pair of Ehrenfest urns with a total of 5 molecules divided between them.a. Find the transition matrix for the Markov chain that models the number of molecules in urn A, and show that this
Consider an unbiased random walk with reflecting boundaries on {1; 2; 3; 4}. Find the communication classes for this Markov chain and determine whether it is reducible or irreducible.
Reorder the states in the Markov chain in Exercise 2 to produce a transition matrix in canonical form.Data From Exercise 2In Exercises 1–6, consider a Markov chain with state space with {1;
Reorder the states in the Markov chain in Exercise 3 to produce a transition matrix in canonical form.Data From Exercise 3Consider a Markov chain with state space with {1; 2,......, n} and the given
Consider an unbiased random walk on the set {1; 2; 3; 4}. What is the probability of moving from 2 to 3 in exactly 3 steps if the walk has a. Reflecting boundaries? b. Absorbing boundaries?
In Exercises 13 and 14, find the transition matrix for the simple random walk on the given graph. 4 5 2 3
In Exercises 13 and 14, consider a simple random walk on the given graph. Show that the Markov chain is irreducible and calculate the mean return times for each state. 1 4 5 2 3
Consider a simple random walk on the following graph.a. Suppose that the walker begins in state 5. What is the expected number of visits to state 2 before the walker visits state 1?b. Suppose again
Consider an unbiased random walk with reflecting boundaries on {1; 2; 3; 4}.a. Find the transition matrix for the Markov chain and show that this matrix is not regular.b. Assuming that the
Consider an unbiased random walk with absorbing boundaries on {1; 2; 3; 4}. Find the communication classes for this Markov chain and determine whether it is reducible or irreducible.
In Exercises 13 and 14, consider a simple random walk on the given graph. In the long run, what fraction of the time will the walk be at each of the various states? 2 3
Consider a simple random walk on the following graph.a. Suppose that the walker begins in state 3. What is the expected number of visits to state 2 before the walker visits state 1?b. Suppose again
In Exercises 13 and 14, consider a simple random walk on the given graph. In the long run, what fraction of the time will the walk be at each of the various states? 4 5 2 3
Consider a biased random walk on the set {1; 2; 3; 4} with probability p = .2 of moving to the left. What is the probability of moving from 2 to 3 in exactly 3 steps if the walk hasa. Reflecting
Consider a biased random walk with reflecting boundaries on {1; 2; 3; 4} with probability p = .2 of moving to the left.a. Find the transition matrix for the Markov chain and show that this matrix is
In Exercises 13 and 14, consider a simple random walk on the given graph. Show that the Markov chain is irreducible and calculate the mean return times for each state. 1 2 5 3
Exercises 14–18 show how the model for run production in the text can be used to determine baseball strategy. Suppose that you are managing a baseball team and have access to the matrices M and SM
Consider the second columns of the matrices M and SM, which correspond to the “Runner on first, none out” state.a. What information does the sum of the second column of M give?b. What value can
Reorder the states in the Markov chain in Exercise 4 to produce a transition matrix in canonical form.Data From Exercise 4Consider a Markov chain with state space with {1; 2,......, n} and the given
Reorder the states in the Markov chain in Exercise 5 to produce a transition matrix in canonical form.Data From Exercise 5Consider a Markov chain with state space with {1; 2,......, n} and the given
In Exercises 13 and 14, find the transition matrix for the simple random walk on the given graph. 4 2 3
In Exercises 15 and 16, consider a simple random walk on the given directed graph. In the long run, what fraction of the time will the walk be at each of the various states? 1 3 نرا 2 4
In Exercises 15 and 16, consider a simple random walk on the given directed graph. Show that the Markov chain is irreducible and calculate the mean return times for each state. 1 3 2 4
In Exercises 15 and 16, consider a simple random walk on the given directed graph. Show that the Markov chain is irreducible and calculate the mean return times for each state. 1 2 3 4 5
Consider a simple random walk on the following directed graph. Suppose that the walker starts at state 1.a. How many visits to state 2 does the walker expect to make before visiting state 3?b. How
The sum of the column of M corresponding to the “Runner on second, none out” state is 4:53933, and the column of SM corresponding to the “Runner on second, none out” state isHow many earned
In Exercises 15 and 16, find the transition matrix for the simple random walk on the given directed graph. 3 2 4
In Exercises 15 and 16, find the transition matrix for the simple random walk on the given directed graph. 1 2 3 4 5
In Exercises 15 and 16, consider a simple random walk on the given directed graph. In the long run, what fraction of the time will the walk be at each of the various states? 1 2 3 4 5
Reorder the states in the Markov chain in Exercise 6 to produce a transition matrix in canonical form.Data From Exercise 6In Exercises 1–6, consider a Markov chain with state space with {1;
Consider a simple random walk on the following directed graph. Suppose that the walker starts at state 4.a. How many visits to state 3 does the walker expect to make before visiting state 2?b. How
The sum of the column of M corresponding to the “Bases empty, one out” state is 3:02622, and the column of SM corresponding to the “Bases empty, one out” state isHow many earned runs do you
Consider the mouse in the following maze from Section 10.1, Exercise 17.The mouse must move into a different room at each time step and is equally likely to leave the room through any of the
In Exercises 17 and 18, suppose a mouse wanders through the given maze. The mouse must move into a different room at each time step and is equally likely to leave the room through any of the
Consider the mouse in the following maze from Section 10.1, Exercise 17.The mouse must move into a different room at each time step and is equally likely to leave the room through any of the
Consider the mouse in the following maze from Section 10.1, Exercise 17.Data From Section 10.1 Exercise 17Suppose a mouse wanders through the given maze. The mouse must move into a different room at
Consider the mouse in the following maze from Section 10.1, Exercise 18.If the mouse starts in room 2, how long on average will it take the mouse to return to room 2?Data From Section 10.1 Exercise
Consider the mouse in the following maze from Section 10.1, Exercise 18.What fraction of the time does it spend in room 3?Data From Section 10.1 Exercise 18The mouse is placed in room 3 of the maze
Find the transition matrix for the Markov chain in Exercise 9 and reorder the states to produce a transition matrix in canonical form.Data From Exercise 9In Exercises 7–10, consider a simple random
Find the transition matrix for the Markov chain in Exercise 10 and reorder the states to produce a transition matrix in canonical form.Data From Exercise 10Consider a simple random walk on the given
Consider the mouse in the following maze, which includes “one-way” doors, from Section 10.1, Exercise 19.Show thatis a steady-state vector for the associated Markov chain, and interpret this
Consider the mouse in the following maze from Section 10.1, Exercise 18.If the mouse starts in room 1, what is the probability that the mouse visits room 3 before visiting room 4?Data From Section
Consider the mouse in the following maze from Section 10.1, Exercise 19.Data From Section 10.1, Exercise 19 19. The mouse is placed in room 1 of the following maze. a. Construct a transition matrix
The mouse is placed in room 3 of the maze shown below.a. Construct a transition matrix and an initial probability vector for the mouse’s travels.b. What are the probabilities that the mouse will be
Suppose that a runner for your team is on first base with no outs. You have to decide whether to tell the baserunner to attempt to steal second base. If the steal is successful, there will be a
Consider the mouse in the following maze from Section 10.1, Exercise 19.Data From Section 10.1 Exercise 19If the mouse starts in room 1, how many steps on average will it take the mouse to get
In Exercises 19 and 20, suppose a mouse wanders through the given maze, some of whose doors are “one-way”: they are just large enough for the mouse to squeeze through in only one direction. The
The mouse is placed in room 1 of the maze shown.a. Construct a transition matrix and an initial probability vector for the mouse’s travels.b. What are the probabilities that the mouse will be in
In the previous exercise, let p be the probability that the baserunner steals second base successfully. For which values of p would you as manager call for an attempted steal?Data from previous
Consider the mouse in the following maze from Section 10.1, Exercise 20.a. Identify the communication classes of this Markov chain as recurrent or transient.b. Find the period of each communication
Consider the mouse in the following maze, which includes “one-way” doors. 1 4 3 2 5
Mark each statement. Justify each answer. In each part, A represents an n x n matrix.(T/F) If A is an orthogonal matrix, then ΙΙAxΙΙ= ΙΙxΙΙ for all x in Rn.
In exercises 3–6, Find(a) The maximum value of Q(x) subject to the constraint xTx = 1, (b) A unit vector u where this maximum is attained, and (c) The maximum of Q(x) subject to the
Find the matrix of the quadratic form. Assume x is in R3.a.b. 5x² + 3x² - 7x² - 4x1x2 + 6X1X3 − 2x2x3
Find the matrix of the quadratic form. Assume x is in R3.a.b. 5x² − 3x² + 7x² + 8x1x2 − 4X1 X3
Repeat Exercise 7 for the data in Exercise 2.Data From Exercise 7Let x1, x2 denote the variables for the two-dimensional data in Exercise 1. Find a new variable y of the form y1 = c1x1 + c2x2,
Then make a change of variable, x = Py that transform the quadratic form into one with no cross-product terms. Write the new quadratic form. Construct P using the methods of Section 7.1. 6x² - 4x1x2
Then make a change of variable, x = Py that transform the quadratic form into one with no cross-product terms. Write the new quadratic form. Construct P using the methods of Section 7.1. 2 3x² +
In Exercises 21–30, matrices are n x n and vectors are in Rn. Mark each statement True or False. Justify each answer.If A is symmetric and P is an orthogonal matrix, then the change of variable x =
If A is m x n, then the matrix G = ATA is called the Gram matrix of A. In this case, the entries of G are the inner products of the columns of A. (See Exercises 25 and 26.)Show that if an n x n
In Exercises 25–32, mark each statement True or False. Justify each answer.(T/F) There are symmetric matrices that are not orthogonally diagonalizable.
Every complex number z can be written in polar form z = r (cosφ + i sinφ) where r is a nonnegative number and cosφ + i sinφ is a complex number of modulus 1.a. Prove that any n x n matrix A
Compute the singular values of the 5 x 5 matrix in Exercise 10 in Section 2.3, and compute the condition number σ1 = σ5.Data From Exercise 10 in Section 2.3Unless otherwise specified, assume
Construct the pseudoinverse of A. Begin by using a matrix program to produce the SVD of A, or, if that is not available, begin with an orthogonal diagonalization of ATA. Use the, pseudoinverse to
Concern an m x n matrix A with a reduced sin- gular value decomposition, A = Ur DVrT, and the pseudoinverse A+ = Vr D–1UTr.Given any b in Rm, adapt Exercise 28 to show that A+b is the
Let x.t / be a cubic Bézier curve determined by points p0, p1, p2, and p3.a. Compute the tangent vector x'(t). Determine how x'(0) and x'(1) are related to the control points, and give geometric
Show that if A is an n x n symmetric matrix, then (Ax) · y = x · (Ay) for all x, y in Rn.
To practice the methods of this section, do not use an eigenvector routine from your matrix program. Instead, use the program to find the eigenvalues, and, for each eigenvalue λ, find an orthonormal
To practice the methods of this section, do not use an eigenvector routine from your matrix program. Instead, use the program to find the eigenvalues, and, for each eigenvalue λ, find an orthonormal
To practice the methods of this section, do not use an eigenvector routine from your matrix program. Instead, use the program to find the eigenvalues, and, for each eigenvalue λ, find an orthonormal
To practice the methods of this section, do not use an eigenvector routine from your matrix program. Instead, use the program to find the eigenvalues, and, for each eigenvalue λ, find an orthonormal
The parametric vector form of a B-spline curve was defined in the Practice Problems aswhere p0, p1, p2, and p3 are the control points.a. Show that for 0 ≤ t ≤ 1(x)t / is in the convex hull of the
In Exercises 3 and 4, compute x3 in two ways: by computing x1 and x2, and by computing P3. P = .3 .7 2] . Xo = хо .8 .2 - .5 .5
In Exercises 1–6, consider a Markov chain with state space {1,2,...........,n} and the given transition matrix. Find the communication classes for each Markov chain, and state whether the Markov
In Exercises 3 and 4, consider a Markov chain on {1; 2; 3} with the given transition matrix P. In each exercise, use two methods to find the probability that, in the long run, the chain is in state
In Exercises 4–6, find the matrix A = limn→∞Sn for the Markov chain with the given transition matrix. Assume that the state space in each case is {1,2,................,n}. If reordering of
In Exercises 4–6, find the matrix A = limn→∞Sn for the Markov chain with the given transition matrix. Assume that the state space in each case is {1,2,................,n}. If reordering of
In Exercises 5 and 6, the transition matrix P for a Markov chain with states 0 and 1 is given. Assume that in each case the chain starts in state 0 at time n = 0. Find the probability that the chain
In Exercises 1–6, consider a Markov chain with state space with {1; 2,......, n} and the given transition matrix. Identify the communication classes for each Markov chain as recurrent or transient,
In Exercises 1–6, justify the transition probabilities for the given initial states.Second and third bases occupied
In Exercises 1–6, justify the transition probabilities for the given initial states.First and third bases occupied
In Exercises 5 and 6, find the matrix to which Pn converges as n increases. P = 1/4 2/3] 3/4 1/3
In Exercises 4–6, find the matrix A = limn→∞Sn for the Markov chain with the given transition matrix. Assume that the state space in each case is {1,2,................,n}. If reordering of
In Exercises 1–6, consider a Markov chain with state space {1, 2,...........,n} and the given transition matrix. Find the communication classes for each Markov chain, and state whether the Markov
In Exercises 1–6, justify the transition probabilities for the given initial states.First, second, and third bases occupied
In Exercises 5 and 6, find the matrix to which Pn converges as n increases. P = 1/4 1/4 1/2 3/5 0 2/5 0 1/3 2/3
In Exercises 1–6, consider a Markov chain with state space with {1; 2,......, n} and the given transition matrix. Identify the communication classes for each Markov chain as recurrent or transient,
In Exercises 21–26, mark each statement True or False. Justify each answer.The probability that the Markov chain starting at state i is eventually absorbed by state j is the (j, i)-element of the
In Exercises 21–26, mark each statement True or False. Justify each answer.Transit times may be computed directly from the entries in the transition matrix.
In Exercises 21–26, mark each statement True or False. Justify each answer.If P is a regular stochastic matrix, then Pn approaches a matrix with equal columns as n increases.
Mark each statement True or False. Justify each answer.If the (i, j)- and (j, i)-entries in Pk are positive for some k, then the states i and j communicate with each other.
In Exercises 21–26, mark each statement True or False. Justify each answer.If two states of a Markov chain have different periods, then the Markov chain is reducible.
In Exercises 21–26, mark each statement True or False. Justify each answer.If {xn} is a Markov chain, then xn+1 must depend only on the transition matrix and xn.
In Exercises 21–26, mark each statement True or False. Justify each answer.The (j, i)-element in the fundamental matrix gives the expected number of visits to state i prior to absorption, starting
In Exercises 21–26, mark each statement True or False. Justify each answer.If its transition matrix is regular, then the steady-state vector gives information on long-run probabilities of the
Mark each statement True or False. Justify each answer.If a Markov chain is reducible, then it cannot have a regular transition matrix.
In Exercises 21–26, mark each statement True or False. Justify each answer.All of the states in an irreducible Markov chain are recurrent.

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