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mathematics
linear algebra and its applications
Linear Algebra And Its Applications 6th Global Edition David Lay, Steven Lay, Judi McDonald - Solutions
In Exercises 21–26, mark each statement True or False. Justify each answer.The transition matrix P may change over time.
In Exercises 21–26, mark each statement True or False. Justify each answer.The sum of the column j of the fundamental matrix M is the expected number of time steps until absorption.
Consider the mouse in the following maze from Section 10.1, Exercise 20.If the mouse starts in room 1, how many steps on average will it take the mouse to get to room 5? 1 4 3 2 5
In Exercises 21–26, mark each statement True or False. Justify each answer.Every stochastic matrix is regular.
Mark each statement True or False. Justify each answer.An irreducible Markov chain must have a regular transition matrix.
In Exercises 21–26, mark each statement True or False. Justify each answer.If state i is recurrent and state i communicates with state j , then state j is also recurrent.
In Exercises 21–26, mark each statement True or False. Justify each answer.The rows of a transition matrix for a Markov chain must sum to 1.
In Exercises 21–26, mark each statement True or False. Justify each answer.Every stochastic matrix has a steady-state vector.
In Exercises 21–26, mark each statement True or False. Justify each answer.The (i,j)-element in the fundamental matrix M is the expected number of visits to the transient state j prior to absorption, starting at the transient state i.
Mark each statement True or False. Justify each answer.If it is possible to go from state i to state j in n steps, where n ≥ 0, then states i and j communicate with each other.
In Exercises 21–26, mark each statement True or False. Justify each answer.If two states i and j are both recurrent, then they must belong to the same communication class.
In Exercises 21–26, mark each statement True or False. Justify each answer.The columns of a transition matrix for a Markov chain must sum to 1.
In Exercises 21–26, mark each statement True or False. Justify each answer.The (i, j)-entry in a transition matrix P gives the probability of a move from state j to state i.
In Exercises 21–26, mark each statement True or False. Justify each answer.Every Markov chain must have at least one transient class.
Mark each statement True or False. Justify each answer.The entries in the steady-state vector are the mean return times for each state.
In Exercises 21–26, mark each statement True or False. Justify each answer. If λ = 1 is an eigenvalue of a matrix P, then P is regular.
In Exercises 21–26, mark each statement True or False. Justify each answer.The (i, j) -entry in P3 gives the probability of a move from state i to state j in exactly three time steps.
In Exercises 21–26, mark each statement True or False. Justify each answer.If A is an m x m substochastic matrix, then the entries in An approach 0 as n increases.
In Exercises 21–26, mark each statement True or False. Justify each answer.Every Markov chain must have exactly one recurrent class.
Mark each statement True or False. Justify each answer.If state i communicates with state j and state j communicates with state k, then state i communicates with state k.
In Exercises 21–26, mark each statement True or False. Justify each answer. If then the entries in q may be interpreted as occupation times. lim Xn = q n→∞
Confirm Theorem 5 for the Markov chain in Exercise 7 by taking powers of the transition matrix.Data From Exercise 7In Exercises 7–10, consider a simple random walk on the given directed graph. Identify the communication classes of this Markov chain as recurrent or transient, and find the period
Suppose that the weather in Charlotte is modeled using the Markov chain in Section 10.1, Exercise 27. About how many days elapse in Charlotte between rainy days?Data From Section 10.1 Exercise 27The weather in Charlotte, North Carolina, can be classified as sunny, cloudy, or rainy on a given day.
Suppose that the weather in Charlotte is modeled using the Markov chain in Section 10.1, Exercise 27. If it is sunny today, what is the probability that the weather will be cloudy before it is rainy?Data From Section 10.1 Exercise 27The weather in Charlotte, North Carolina, can be classified as
Confirm Theorem 5 for the Markov chain in Exercise 8 by taking high powers of the transition matrix.Data From Exercise 8Consider a simple random walk on the given directed graph. Identify the communication classes of this Markov chain as recurrent or transient, and find the period of each
In Exercises 29 and 30, consider a set of webpages hyperlinked by the given directed graph. Find the Google matrix for each graph and compute the PageRank of each page in the set. 1 3 2 4 5
Suppose that the weather in Charlotte is modeled using the Markov chain in Section 10.1, Exercise 28. Over the course of a year, about how many days in Charlotte are rainy according to the model?Data From Section 10.1 Exercise 28Suppose that whether it rains in Charlotte tomorrow depends on the
Consider a set of webpages hyperlinked by the given directed graph that was studied in Section 10.2, Exercise 29.If a random surfer starts on page 1, how many mouse clicks on average will the surfer make before becoming stuck at a dangling node?Data From Section 10.2 Exercise 29Consider a set
Suppose that the weather in Charlotte is modeled using the Markov chain in Section 10.1, Exercise 28. If it rained yesterday and today, how many days on average will it take before there are two consecutive days with no rain?Data From Section 10.1 Exercise 28Suppose that whether it rains in
Suppose that whether it rains in Charlotte tomorrow depends on the weather conditions for today and yesterday. Climate data from 2003 show thatIf it rained yesterday and today, then it will rain tomorrow with probability .58.If it rained yesterday but not today, then it will rain tomorrow with
The following set of webpages hyperlinked by the directed graph was studied in Section 10.2, Exercise 29.Data From Section 10.2 Exercise 29Consider a set of webpages hyperlinked by the given directed graph. Find the Google matrix for each graph and compute the PageRank of each page in the set.
Consider a set of webpages hyperlinked by the given directed graph that was studied in Section 10.2, Exercise 30.If a random surfer starts on page 3, what is the probability that the surfer will eventually become stuck on page 1, which is a dangling node?Data From Section 10.2 Exercise 30Consider a
Consider a set of four webpages hyperlinked by the directed graph in Exercise 15. If a random surfer starts at page 1, what is the probability that the surfer will be at each of the pages after 3 clicks?Data From Exercise 15Find the transition matrix for the simple random walk on the given directed
In Exercises 29 and 30, consider a set of webpages hyperlinked by the given directed graph. Find the Google matrix for each graph and compute the PageRank of each page in the set. 2 3 4 5 6
Suppose that the weather in Charlotte is modeled using the Markov chain in Section 10.1, Exercise 28. About how many days elapse in Charlotte between consecutive rainy days?Data From Section 10.1 Exercise 28Suppose that whether it rains in Charlotte tomorrow depends on the weather conditions for
Follow the plan of Exercise 29 to confirm Theorem 5 for the Markov chain with transition matrixData From Exercise 29 Consider the Markov chain on {1; 2; 3} with transition matrixData From Theorem 5 P = 0 1 Lo 1/2 0 1/2 1 0
Consider a set of five webpages hyperlinked by the directed graph in Exercise 16. If a random surfer starts at page 2, what is the probability that the surfer will be at each of the pages after 4 clicks?Data From Exercise 16In Exercises 15 and 16, find the transition matrix for the simple random
Confirm Theorem 5 for the Markov chain in Example 6. EXAMPLE 6 A Markov chain on {1, 2, 3} has transition matrix P= = 1 0 1 0 2 3 0 1 0 0 1 0 1 2 3
Consider the pair of Ehrenfest urns studied in Section 10.2, Exercise 9. Suppose that there are now 2 molecules in urn A. How many steps on average will be needed until there are again 2 molecules in urn A?Data From Section 10.2 Exercise 9Consider a pair of Ehrenfest urns with a total of 4
Exercises 31–34 concern the Markov chain model for scoring a tennis match described in Section 10.1, Exercise 35. Suppose that players A and B are playing a tennis match, that the probability that player A wins any point is p D :6, and that the game is currently at “deuce.”How many more
Consider a model for signal transmission in which data is sent as two-bit bytes. Then there are four possible bytes, 00, 01, 10, and 11, which are the states of the Markov chain. At each stage there is a probability p that each bit will pass through the stage unchanged.
Consider the pair of Ehrenfest urns studied in Section 10.2, Exercise 10. Suppose that urn A is now empty. How many steps on average will be needed until urn A is again empty?Data From Section 10.2 Exercise 10Consider a pair of Ehrenfest urns with a total of 5 molecules divided between them.a. Find
Let 0 ≤ q ≤ 1. Show thatis a steady-state vector for the Markov chain in Example 3.Data From Example 3 q 0 0 0 1-q
Consider beginning with an individual of known type and mating it with a hybrid, then mating an offspring of this mating with a hybrid, and so on. At each step, an offspring is mated with a hybrid. The type of the offspring can be modeled by a Markov chain with states AA, Aa, and aa.a. Find the
Exercises 31–34 concern the Markov chain model for scoring a tennis match described in Section 10.1, Exercise 35. Suppose that players A and B are playing a tennis match, that the probability that player A wins any point is p D :6, and that the game is currently at “deuce.”Find the
Consider a model for signal transmission in which data is sent as three-bit bytes. Construct the transition matrix for the model.
Another version of the Ehrenfest model for diffusion starts with k molecules of gas in each urn. One of the 2k molecules is picked at random just as in the Ehrenfest model in the text. The chosen molecule is then moved to the other urn with a fixed probability p and is placed back in its urn with
Consider the variation of the Ehrenfest urn model of diffusion studied in Section 10.1, Exercise 33, where one of the 2k molecules is chosen at random and is then moved between the urns with a fixed probability p.a. Let k = 3 and suppose that p = 1/2. Show that the transition matrix for the Markov
A variation of the Ehrenfest model of diffusion was studied in Section 10.2, Exercise 33. Consider this model with k = 3 and p = 1/2 and suppose that there are now 3 molecules in urn A. How many draws on average will be needed until there are again 3 molecules in urn A?Data From Section 10.2
Another model for diffusion is called the Bernoulli-Laplace model. Two urns (urn A and urn B) contain a total of 2k molecules. In this case, k of the molecules are of one type (called type I molecules) and k are of another type (type II molecules). In addition, k molecules must be in each urn at
Consider the Bernoulli-Laplace model of diffusion studied in Section 10.2, Exercise 34. Let k = 5. Suppose that all of the type I molecules are now in urn A. How many draws on average will be needed until all of the type I molecules are again in urn A?Data From Section 10.2 Exercise 34Consider the
Consider the Bernoulli-Laplace diffusion model studied in Section 10.1, Exercise 34.a. Let k = 5 and show that the transition matrix for the Markov chain that models the number of type I molecules in urn A is regular.b. Let k = 5. In what state will this chain spend the most steps, and what
To win a game in tennis, one player must score four points and must also score at least two points more than his or her opponent. Thus if the two players have scored an equal number of points (four or more), which is called “deuce” in tennis jargon, one player must then score two points in a
A Markov chain model for scoring a tennis game was studied in Section 10.1, Exercise 35. What are the communication classes for this Markov chain?Data From Section 10.1 Exercise 35To win a game in tennis, one player must score four points and must also score at least two points more than his or her
Exercises 35–40 concern the two Markov chain models for scoring volleyball games described in Section 10.1, Exercise 36. Suppose that teams A and B are playing a 15-point volleyball game that is tied 15-15 with team A serving. Suppose that the probability p that team A wins any rally for which it
Volleyball uses two different scoring systems in which a team must win by at least two points. In both systems, a rally begins with a serve by one of the teams and ends when the ball goes out of play or touches the floor or a player commits a fault. The team that wins the rally gets to serve for
The weather in Charlotte, North Carolina, can be classified as sunny, cloudy, or rainy on a given day. Climate data from 2003 reveal thatIf a day is sunny, then the next day will be sunny with probability .65, cloudy with probability .1, and rainy with probability .25.If a day is cloudy, then the
Suppose that the weather in Charlotte is modeled using the Markov chain in Section 10.1, Exercise 27. Over the course of a year, about how many days in Charlotte are sunny, cloudy, and rainy according to the model?Data From Section 10.1 Exercise 27The weather in Charlotte, North Carolina, can be
In Exercises 37 and 38, consider the Markov chain on {1; 2; 3; 4; 5} with transition matrixShow that this Markov chain is irreducible. 0 1/3 P = | 2/3 0 0 0 0 0 2/5 3/5 0 0 0 1/5 4/5 1/2 0 0 1/2 0 0 0 0 0
A Markov chain model for the rally point method for scoring a volleyball game was studied in Section 10.1, Exercise 36. What are the communication classes for this Markov chain?Data From Section 10.1 Exercise 36Volleyball uses two different scoring systems in which a team must win by at least two
Consider the Markov chain in Example 4.a. Show thatis a steady-state vector for this Markov chain.b. Compute the average of the entries in P20 and P21 given in Example 4. What do you find? 1/8 1/4 1/4 1/4 1/8
n Exercises 37 and 38, consider the Markov chain on {1; 2; 3; 4; 5} with transition matrixSuppose the chain starts in state 1. What is the expected number of steps until it is in state 1 again? 0 1/3 P = | 2/3 0 0 0 0 0 2/5 3/5 0 0 0 1/5 4/5 1/2 0 0 1/2 0 0 0 0 0
Let 0 ≤ p, q ≤ 1, and definea. Show that 1 and p + q – 1 are eigenvalues of P.b. By Theorem 1, for what values of p and q will P fail to be regular?c. Find a steady-state vector for P.Data From Section 10.2 Theorem 1 P=[1²p 1-q 9
Let 0 ≤ p, q ≤ 1, and definea. For what values of p and q is P a regular stochastic matrix?b. Given that P is regular, find a steady-state vector for P. P = P 9 1-p-q 9 1-p-q P 1-p-q P 9
Exercises 35–40 concern the two Markov chain models for scoring volleyball games described in Section 10.1, Exercise 36. Suppose that teams A and B are playing a 15-point volleyball game that is tied 15-15 with team A serving. Suppose that the probability p that team A wins any rally for which it
Suppose that P is a stochastic matrix all of whose entries are greater than or equal to p. Show that all of the entries in Pn are greater than or equal to p for n = 1; 2,........
Let A be an m x m stochastic matrix, let x be in Rm, and let y = Ax. Show thatwith equality holding if and only if all of the nonzero entries in x have the same sign. |y₁| + ··· + [ym] ≤ |x₁| + ··· + |xm|
Exercises 35–40 concern the two Markov chain models for scoring volleyball games described in Section 10.1, Exercise 36. Suppose that teams A and B are playing a 15-point volleyball game that is tied 15-15 with team A serving. Suppose that the probability p that team A wins any rally for which it
Consider a Markov chain on {1,2,3,4,5} with transition matrix a. What are the recurrent and transient classes for this chain?b. Find the limiting matrix for each recurrent class.c. Determine the long-range probabilities for the Markov chain starting from each transient state.d. Use the results
Exercises 35–40 concern the two Markov chain models for scoring volleyball games described in Section 10.1, Exercise 36. Suppose that teams A and B are playing a 15-point volleyball game that is tied 15-15 with team A serving. Suppose that the probability p that team A wins any rally for which it
How does the presence of dangling nodes in a set of hyperlinked webpages affect the communication classes of the associated Markov chain?
Exercises 35–40 concern the two Markov chain models for scoring volleyball games described in Section 10.1, Exercise 36. Suppose that teams A and B are playing a 15-point volleyball game that is tied 15-15 with team A serving. Suppose that the probability p that team A wins any rally for which it
Show that the communication relation is transitive. Show that the (i, k)-entry of Pn+m must be greater than or equal to the product of the (i; j)-entry of Pm and the (j, k) entry of Pn.
Consider a Markov chain on {1,2,3,4,5,6} with transition matrix a. What are the recurrent and transient classes for this chain?b. Find the limiting matrix for each recurrent class.c. Find the absorption probabilities from each transient state into each recurrent class.d. Use the results of
Mark each statement. Justify each answer.(T/F) Similar matrices always have exactly the same eigenvalues.
Use the power method with the x0 given. List {xk} and {μk} for k = 1,...,5. 2 4= [² ] - [8] A 1 5 Xo 4
Mark each statement. Justify each answer.(T/F) Eigenvectors must be nonzero vectors.
Let H be the hyperplane through the listed points. (a) Find a vector n that is normal to the hyperplane. (b) Find a linear functional f and a real number d such that H = [f : d]. 1 22
In Exercises 1 and 2, set up the initial simplex tableau for the given linear programming problem. Maximize subject to 21x₁ + 25x2 + 15x3 2x17x₂ + 10x3 ≤ 20 3x14x2 18x3 ≤ 25 and x₁ ≥ 0, x₂ ≥ 0, X3 ≥ 0.
A quartic Bézier curve is determined by five control points, p0, p1, p2, p3, and p4:Construct the quartic basis matrix MB for x(t). x(t) = (1 t) Po + 4t (1 – t)³p₁ + 6t²(1 – t)²p₂ + 4t³(1-1)P3 + 14p4 for 0 ≤t≤1
Suppose that the solutions of an equation Ax = b are all of the form x = x3 u + p, where Find points v1 and v2 such that the solution set of Ax = b is aff {v1; v2}. -2 and p = -3]. 0
Determine if the set of points is affinely dependent. If so, construct an affine dependence relation for the points. -3 [] [] [22] 4 -2
Write y as an affine combination of the other points listed, if possible. ~=[{}]×2=[¯1]·»=[2] = [5] V2 y V3
Determine if the set of points is affinely dependent. If so, construct an affine dependence relation for the points. 2 [HD] -4 8 2 0 15 -9
In Exercises 1–24, mark each statement True or False. Justify each answer.Every payoff matrix has at least one saddle point.
Let L be the line in R2 through the points Find a linear functional f and a real number d such that L = [f : d]. [4] and [3] -1
Mark each statement True or False. Justify each answer.The barycentric coordinates of a point in R2 are always nonnegative.
Mark each statement True or False. Justify each answer.If v1,..., vp are in Rn and if the set {v1 v2, v3 – v2, vp - v2} is linearly dependent, then {v1,..., vp} is affinely dependent.
Suppose a Bézier curve is translated to x(t) + b. That is, for 0 ≤ t ≤ 1, the new curve isShow that this new curve is again a Bézier curve. x(t) = (1-1)³ po + 3t (1 - 1)²p₁ +31² (1-1) p₂ + 1³p3 + b
Write y as an affine combination of the other points listed, if possible. * = [2]·2= [-2] - [8] = []=[] V2 V3 V4 y
In Exercises 1–24, mark each statement True or False. Justify each answer.A negative entry aij in a payoff matrix indicates the amount player R has to pay player C when R choses action i and C choses action j .
Let S = {(x, y) : x2 + (y – 1)2 ≤ 1} ∪ {(3, 0)}. Is the origin an extreme point of conv S? Is the origin a vertex of conv S?
Let L be the line in R2 through the points Find a linear functional f and a real number d such that L = [f : d]. [4] and 3 [³]
Mark each statement True or False. Justify each answer.Every subspace is an affine set.
Mark each statement True or False. Justify each answer.Every subspace is a flat.
Mark each statement True or False. Justify each answer.Two flats are parallel if their intersection is empty.
Mark each statement True or False. Justify each answer.A hyperplane is a 4-dimensional flat.
Mark each statement True or False. Justify each answer.The affine hull of two points v1 and v2 is the set of all points y = t v1 + (1 – t)v2, with t in R.
In mark each statement. Justify each answer.(T/F) For a nonzero v in Rn, the matrix vvT is called a projection matrix.
Mark each statement True or False. Justify each answer.Given v1, v2,..................,vp in Rn and scalars c1,..........,cp, an affine combination of v1, v2,........., vp is a linear combination c1v1 +..........+cpvp such that the weights satisfy c1 +..........+cp = 1.
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