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linear algebra and its applications

Questions and Answers of Linear Algebra And Its Applications

(T/F) If each vector ej in the standard basis for Rn is an eigenvector of A, then A is a diagonal matrix.
P2 with the inner product given by evaluation at –1, 0, and 1.Compute ΙΙpΙΙ and ΙΙqΙΙ, for p and q in Exercise 3.Data From Exercise 3P2 with the inner product given by evaluation at –1,
Refer to vectors in Rn (or Rm) with the standard inner product. Justify each answer.(T/F) If r is any scalar, then ΙΙrvΙΙ = r ΙΙvΙΙ.
Refer to vectors in Rn (or Rm) with the standard inner product. Justify each answer.(T/F) If a vector y coincides with its orthogonal projection onto a subspace W, then y is in W.
In Exercises 19–24, u; v, and w are vectors. Mark each statement True or False. Justify each answer.If (u, u) = 0, then u = 0.
Find the steady-state vector. .7 0 .3 ୯ .6 .4
Use Example 6 to list the eigenvalues of A. In each case, the transformation x ↦ Ax is the composition of a rotation and a scaling. Give the angle φ of the rotation, where -π < φ ≤ π, and
Define T : R2 → R2 by T(x) = Ax. Find a basis B for R2 with the property that [T]B is diagonal. A 4 =[-1 -2 3
Use Example 6 to list the eigenvalues of A. In each case, the transformation x ↦ Ax is the composition of a rotation and a scaling. Give the angle φ of the rotation, where -π < φ ≤ π, and
Use Example 6 to list the eigenvalues of A. In each case, the transformation x ↦ Ax is the composition of a rotation and a scaling. Give the angle φ of the rotation, where -π < φ ≤ π, and
Use a matrix program to find the eigenvalues of the matrix. Then use the method of Example 4 with a row reduction routine to produce a basis for each eigenspace.Data from in Example 4 9 -4 -2 -56 32
Use a matrix program to find the eigenvalues of the matrix. Then use the method of Example 4 with a row reduction routine to produce a basis for each eigenspace.Data from in Example 4 4 -7 5 -2 -3 -9
Use a matrix program to find the eigenvalues of the matrix. Then use the method of Example 4 with a row reduction routine to produce a basis for each eigenspace.Data from in Example 4 -4 -4 14 20
a. Find the steady-state vector for the Markov chain in Exercise 3.b. What is the probability that after many days a specific student is ill? Does it matter if that person is ill today?Data from in
A is an n × n matrix. Justify each answer.(T/F) A number c is an eigenvalue of A if and only if the equation (A - cI)x = 0 has a nontrivial solution.
Letand Show that the least-squares line for the data.(x1; y1).......(xn,yn) must pass through (x̅,y̅). That is, show that x̅ and y̅ satisfy the linear equation Derive this equation from the
P2 with the inner product given by evaluation at –1, 0, and 1.Compute (p, q), where p(t) = 4 + t, q(t) = 5 – 4t2.
Derive the normal equations (7) from the matrix form given in this section.Data From Section 6.6 Equation (7) ηβο + β, Σx = Σ βο Σ.x + βι Σ. = Συν
Exercises 27–30 refer to V = C[0, 1], with the inner product given by an integral, as in Example 7.Let V be the space C[-2, 2] with the inner product of Example 7. Find an orthogonal basis for the
P2 with the inner product given by evaluation at –1, 0, and 1.Compute (p, q), where p(t) = 4t - 3t2, q(t) = 1 + 9t2.
Prove Theorem 7. Let U be an m x n matrix with orthonormal columns, and let x and y be in R". Then a. ||UX|| = |x|| b. (Ux).(Uy)=xy c. (Ux). (Uy) = 0 if and only if x.y = 0
Refer to vectors in Rn (or Rm) with the standard inner product. Justify each answer.(T/F) The distance between u and v is ΙΙu - vΙΙ.
In Exercises 19–24, u; v, and w are vectors. Mark each statement True or False. Justify each answer.(u + v, w) = (w, u) + (w, v).
In Exercises 19–24, u; v, and w are vectors. Mark each statement True or False. Justify each answer.|(u, v)| ≤ ||u|| ||v||.
In Exercises 19–24, u; v, and w are vectors. Mark each statement True or False. Justify each answer.(cu, cv) = c(u, v).
In Exercises 19–24, u; v, and w are vectors. Mark each statement True or False. Justify each answer.|(u, u)| = (u, u).
Let u = (u1, u2, u3). Explain why u • u ≥ 0. When is u • u = 0?
Let A be an m x n matrix, and let B be an n x p matrix such that AB = 0. Show that rank A + rank B ≤ n.
According to Theorem 12, a linearly independent set {v1,...,vk in Rn can be expanded to a basis for Rn. One way to do this is to create A = [v1 ... vk e1 ... en], with e1,...,en the columns of
Let B = {1, cost, cos2 t,..., cos6 t and C = {1, cos t, cos 2t,...,cos 6t}. Assume the following trigonometric identities (see Exercise 45 in Section 4.1).Let H be the subspace of functions spanned
Let A = PDP-1 and compute A4. P=[ Р 1 3 3].D= [8 2 5
Let A = PDP-1 and compute A4. 0 [59]- [ ] -- a E- T d
Diagonalize the matrices, if possible. The eigenvalues are as follows: (11)λ = 1, 2, 3; (12)λ = 1, 4; (13)λ = 5, 1; (14)λ = 3, 4; (15)λ = 3, 1; (16)λ = 2, 1. [59] -
Use Example 6 to list the eigenvalues of A. In each case, the transformation x ↦ Ax is the composition of a rotation and a scaling. Give the angle φ of the rotation, where -π < φ ≤ π, and
Find the B-matrix for the transformation x ↦ Ax, when B = {b1, b2}. 3 A = [ 2² ] = [2] = [] А b₁ ,b₂ -2 3
Find the characteristic polynomial and the eigenvalues of the matrices. [2 7-2 3
Diagonalize the matrices, if possible. The eigenvalues are as follows: (11) λ = 1, 2, 3; (12)λ = 1, 4; (13)λ = 5, 1; (14)λ = 3, 4; (15)λ = 3, 1; (16)λ = 2, 1. 5 0 1 5
Is λ = 3 an eigenvalue ofIf so, find one corresponding eigenvector. 1 2 3-2 0 1 2 1? 1
Use Example 6 to list the eigenvalues of A. In each case, the transformation x ↦ Ax is the composition of a rotation and a scaling. Give the angle φ of the rotation, where -π < φ ≤ π, and
Use Example 6 to list the eigenvalues of A. In each case, the transformation x ↦ Ax is the composition of a rotation and a scaling. Give the angle φ of the rotation, where -π < φ ≤ π, and
First change the given pattern into a vector w of zeros and ones and then use the method illustrated in Example 5 to find a matrix M so that wT M w = 0, but uT M u ≠ 0 for all other nonzero vectors
First change the given pattern into a vector w of zeros and ones and then use the method illustrated in Example 5 to find a matrix M so that wT M w = 0, but uT M u ≠ 0 for all other nonzero vectors
First change the given pattern into a vector w of zeros and ones and then use the method illustrated in Example 5 to find a matrix M so that wT M w = 0, but uT M u ≠ 0 for all other nonzero vectors
Let U be a square matrix with orthonormal columns. Explain why U is invertible.
Example 2 in Section 4.8 displayed a low-pass linear filter that changed a signal {yk} into {yk+1} and changed a higher-frequency signal {ωk} into the zero signal, where yk = cos(πk/4) and (ωk) =
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,
Let U be the 8 × 4 matrix in Exercise 43 in Section 6.2. Find the closest point to y = (1, 1, 1, 1, 1, 1, 1, 1) in Col U. Write the keystrokes or commands you use to solve this problem.Data from in
In 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 of each
In 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 of each
Consider the mouse in the following maze from Section 10.1, Exercise 17.If the mouse starts in room 3, how long on average will it take the mouse to return to room 3?Data from Section 10.1, Exercise
A genetic trait is often governed by a pair of genes, one inherited from each parent. The genes may be of two types, often labeled A and a. An individual then may have three different pairs: AA, Aa
Let x(t) be the B-spline in Exercise 2, with control points p0, p1, p2, and p3.a. Compute the tangent vector x'(t) and determine how the derivatives x'(0) and x'(1) are related to the control points.
Let and S = {b1; b2; b3}. ------ b₁ [ , b₂ = 0, b3 -5
Let and S = {b1; b2; b3}. ------ b₁ [ , b₂ = 0, b3 -5
Repeat Exercise 7 whenData From Exercise 7Let and S = {v1; v2; v3}. It can be shown that S is linearly independent.a. Is p1 in Span S? Is p1 in aff S?b. Is p2 in Span S? Is p2 in
A B-spline is built out of B-spline segments, described in Exercise 2. Let p0,................,p4 be control points. For 0 ≤ t ≤ 1, let x(t) and y(t) be determined by the geometry matrices
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 and 2, determine whether P is a stochastic matrix. If P is not a stochastic matrix, explain why not. .3 a. P = P=[4] • P=[23] b. .6 .4 .6
Let x(t) and y(t) be cubic Bézier curves with control points (p0, p1,p2,p3) and (p3, p4, p5, p6), respectively, so that x(t) and y(t) are joined at p3. The following questions refer to the
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 and 2, determine whether P is a stochastic matrix. If P is not a stochastic matrix, explain why not. .3 .4 .3 .7 a. P = P=[₂4] 6. P=[22] b. .7 .6 .4 .6
In Exercises 1–3, find the fundamental matrix of the Markov chain with the given transition matrix. Assume that the state space in each case is {1, 2,............,n}. If reordering of states is
In Exercises 1 and 2, consider a Markov chain on {1; 2} 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 1.
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 and 2, consider a Markov chain on {1; 2} 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 1.
Exercises 9–16 relate to a primal linear programming problem of finding x in Rn so as to maximize f (x) = cTx subject to Ax ≤ b and x ≥ 0. Mark each statement True or False (T/F). Justify each
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 1–3, find the fundamental matrix of the Markov chain with the given transition matrix. Assume that the state space in each case is {1, 2,............,n}. If reordering of states is
In Exercises 1–6, justify the transition probabilities for the given initial states.First base occupied
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, 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, compute x3 in two ways: by computing x1 and x2, and by computing P3. P .6 .5 = [$_$], x= [6] .4 .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 1–3, find the fundamental matrix of the Markov chain with the given transition matrix. Assume that the state space in each case is {1, 2,............,n}. If reordering of states is
In Exercises 1–6, justify the transition probabilities for the given initial states.Third base occupied
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 1–6, justify the transition probabilities for the given initial states.First and second bases occupied
Suppose that the Markov chain in Exercise 1 starts at state 3. How many visits will the chain make to state 4 on average before absorption?Data From Exercise 1In Exercises 1–3, find the fundamental
Consider the mouse in the following maze from Section 10.1, Exercise 19.Find the communication classes for the Markov chain that models the mouse’s travels through this maze. Is this Markov chain
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 7 and 8, determine whether the given matrix is regular. Explain your answer. 1/3 P = 1/3 1/3 0 1/2 1/2 1/2 1/2 0
Suppose that the Markov chain in Exercise 2 starts at state 4. How many steps will the chain take on average before absorption?Data From Exercise 2In Exercises 1–3, find the fundamental matrix of
Consider the mouse in the following maze from Section 10.1, Exercise 20.Find the communication classes for the Markov chain that models the mouse’s travels through this maze. Is this Markov chain
Find the complete transition matrix for the model using the Major League data in Table 5. TABLE 5 Major League Batting Statistics-2006 Season Walks Hit Batsman Singles Doubles Triples Home
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 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 of each
In Exercises 9 and 10, consider the set of webpages hyperlinked by the given directed graph. Find the communication classes for the Markov chain that models a random surfer’s progress through this
In Exercises 7 and 8, determine whether the given matrix is regular. Explain your answer. P = [1/2] 0 1/2 0 0 2/5 0 3/5 1/3 0 2/3 0 0 3/7 0 4/7
Suppose that the Markov chain in Exercise 3 starts at state 1. How many steps will the chain take on average before absorption?Data From Exercise 3In Exercises 1–3, find the fundamental matrix of
It can be shown that the sum of the first column of M for the 2006 Major League data is 4:53933, and that the first column of SM for the 2006 Major League data is Find the expected number of
In 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 of each
Suppose that the Markov chain in Exercise 4 starts at state 3. What is the probability that the chain is absorbed at state 1?Data From Exercise 4In Exercises 4–6, find the matrix A = limn→∞Sn
In Exercises 9 and 10, consider the set of webpages hyperlinked by the given directed graph. Find the communication classes for the Markov chain that models a random surfer’s progress through this
Suppose that the Markov chain in Exercise 5 starts at state 4. Find the probabilities that the chain is absorbed at states 1, 2, and 3.Data From Exercise 5In Exercises 4–6, find the matrix A =
Consider a pair of Ehrenfest urns labeled A and B. There are currently 3 molecules in urn A and 1 in urn B. What is the probability that the exact same situation will apply after a. 4
Batting statistics for three of the greatest batters in Major League history are shown in Table 6. Compute the transition probabilities for this data for each player. TABLE 6 Batting Statistics for
Consider a pair of Ehrenfest urns with a total of 4 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
Suppose that the Markov chain in Exercise 6 starts at state 5. Find the probabilities that the chain is absorbed at states 2 and 4.Data From Exercise 6In Exercises 4–6, find the matrix A =
Reorder the states in the Markov chain in Exercise 1 to produce a transition matrix in canonical form.Data From Exercise 1In Exercises 1–6, consider a Markov chain with state space with {1;

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