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linear algebra and its applications

Questions and Answers of Linear Algebra And Its Applications

Assume that any initial vector x0 has an eigenvector decomposition such that the coefficient c1 in equation (1) of this section is positive.The tawny owl is a widespread breeding species in Europe
Solve the initial value problem x'(t) = Ax(t) for t ≥ 0, with x(0) = (3,2). Classify the nature of the origin as an attractor, repeller, or saddle point of the dynamical system described by x' =
Isan eigenvector ofIf so, find the eigenvalue. 4 -3 1
Find the characteristic polynomial and the eigenvalues of the matrices. 2 I- 1 4
The matrix A is factored in the form PDP-1. Use the Diagonalization Theorem to find the eigenvalues of A and a basis for each eigenspace. 2 1 1 2 1 32 1 = 2 1 1 2 0 -1 1 -1 0 5 0 0 0 0 1/4 0
Let each matrix in act on C2. Find the eigenvalues and a basis for each eigenspace in C2. 0 -8 4
Justify each answer.(T/F) Each eigenvector of A is also an eigenvector of A2.
LetRepeat Exercise 5, using the following sequence x, Ax,......,A5x.Data from in Exercise 5LetFind a vector with a 1 in the second entry that is close to an eigenvector of A. Use four decimal places.
Solve the initial value problem x'(t) = Ax(t) for t ≥ 0, with x(0) = (3,2). Classify the nature of the origin as an attractor, repeller, or saddle point of the dynamical system described by x' =
Find the characteristic polynomial and the eigenvalues of the matrices. 1 4 -4 6
Find the steady-state vector. i ir in
Let B = {b1, b2, b3} be a basis for a vector space V. Find T(2b1 - b2 + 4b3) when T is a linear transformation from V to V whose matrix relative to B is [T]B -[i = 0-6 1 0 5 -1 1-2 7
Let each matrix in act on C2. Find the eigenvalues and a basis for each eigenspace in C2. 4 -3 3 4
Use the power method with the x0 given. List {xk} and {μk} for k = 1,...,5. S [i] = [2 8] => 0x V L 9
The matrix A is factored in the form PDP-1. Use the Diagonalization Theorem to find the eigenvalues of A and a basis for each eigenspace. 7 6 025 TNO -1 1 120 1 2 3 = 2 1 0 0
Justify each answer.(T/F) Each eigenvector of an invertible matrix A is also an eigenvector of A-1.
Determine ifis a regular stochastic matrix. P -[i 0 71 1.3
Use the power method with the x0 given. List {xk} and {μk} for k = 1,...,5. List μ5 and μ6. A = 1 0 2-2 1 9 1 9 ΧΟ = [] 0 0
Construct the general solution of x' = Ax involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories. A || -3 - 1 2 -1
Find a basis for the eigenspace corresponding to each listed eigenvalue. A 9 [² 2 3 ,λ = 3,9
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. 3 -1 5
Find the characteristic polynomial of each matrix using expansion across a row or down a column. 1 2 0 0 -1 3-1 6 0
Mark each statement. Justify each answer.(T/F) Two eigenvectors corresponding to the same eigenvalue are always linearly dependent.
Determine ifis a regular stochastic matrix. P || .7 .3 0 1
Use the power method with the x0 given. List {xk} and {μk} for k = 1,...,5. List μ5 and μ6. A = 8 1 0 0 -2 3 12 1, Xo 0 -[3] 0 0
Find a basis for the eigenspace corresponding to each listed eigenvalue. A = 14 -4 16-2 , λ = 6
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. 3 4 6 1
Define T : R2 → R2 by T(x) = Ax. Find a basis B for R2 with the property that [T]B is diagonal. = [₁ A = 5 -7 -3 1
Find the characteristic polynomial of each matrix using expansion across a row or down a column. 0 3 0 1 2 3 نا 1 2 0
Another estimate can be made for an eigenvalue when an approximate eigenvector is available. Observe that if Ax = λx, then xTAX = xT(λx) = λ(xTx), and the Rayleigh quotientequals λ. If x is
Construct the general solution of x' = Ax involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories. =[₁ A = 3. -2 1]
Classify the origin as an attractor, repeller, or saddle point of the dynamical system xk+1 = Axk. Find the directions of greatest attraction and/or repulsion. A || .3 -.3 در .4 1.1
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. -1 انکا
Find the characteristic polynomial of each matrix using expansion across a row or down a column. 6 5 1 0 4 0 0 32
Mark each statement. Justify each answer.(T/F) Similar matrices always have exactly the same eigenvectors.
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. 3 -1 - 1 -1 -1 3 -1 -1 -1 -1 3
Classify the origin as an attractor, repeller, or saddle point of the dynamical system xk+1 = Axk. Find the directions of greatest attraction and/or repulsion. A = [₁ .5 -.3 .6 1.4
Construct the general solution of x' = Ax involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories. A -7 -4 10 5
Find the characteristic polynomial of each matrix using expansion across a row or down a column. 6-2 9 5 8 -2 * 0 0 3
Another estimate can be made for an eigenvalue when an approximate eigenvector is available. Observe that if Ax = λx, then xTAX = xT(λx) = λ(xTx), and the Rayleigh quotientequals λ. If x is
Define T : R2 → R2 by T(x) = Ax. Find a basis B for R2 with the property that [T]B is diagonal. A = [ + ] = 9-
Find a basis for the eigenspace corresponding to each listed eigenvalue. 1 4-[32]^- A ,λ = -2,5
Find the characteristic polynomial of each matrix using expansion across a row or down a column. 10 6 0 -3 0 1 1 4
Construct the general solution of x' = Ax involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories. A = -3 -9 2 3
Classify the origin as an attractor, repeller, or saddle point of the dynamical system xk+1 = Axk. Find the directions of greatest attraction and/or repulsion. .4 -.4 A = = [₁ .5 1.3
Mark each statement. Justify each answer.(T/F) The eigenvalues of an upper triangular matrix A are exactly the nonzero entries on the diagonal of A.
Find a basis for the eigenspace corresponding to each listed eigenvalue. A 4-2 3].» 9 -3 λ = 10
Mark each statement. Justify each answer.(T/F) The sum of two eigenvectors of a matrix A is also an eigenvector of A.
Refer to Exercise 2. Which food will the animal prefer after many trials?Data from in Exercise 2A laboratory animal may eat any one of three foods each day. Laboratory records show that if the animal
LetDefine T : R2 → R2 by T(x) = Ax.a. Verify that b1 is an eigenvector of A but A is not diagonalizable.b. Find the B matrix for T. A 4 = [-1 5 Ja | and B5 = {by, by), for by = [¦]. b₂ =
Construct the general solution of x' = Ax involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories. A = -6 -11 -11 2 5 -4 -5 16 -4 10
Find an invertible matrix P and a matrix a C of the form such that the given matrix has the b a form A = PCP-1. D b -b a
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. 0 -4-6 -1 0-3 12 5
Letis an eigenvector for A, and two eigenvalues are .5 and .2. Construct the solution of the dynamical system xk+1 = Axk that satisfies x0 = (0, .3, .7). What happens to xk as k → ∞? A
Find a basis for the eigenspace corresponding to each listed eigenvalue. Α = = 3 1 0 0 0 3 1 0 2 1 1 0 0 0 0 4 λ = 4
For the matrices list the eigenvalues, repeated according to their multiplicities. 0 0 8-4 0 in o 58 0 7 1 -5 1 2 0 0 0 1
Construct the general solution of x' = Ax involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories. A = -8 -12 1 12 827 2 -6 2 5
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. -7 -2 2 24 7 -6 -16 -4 5
Find a basis for the eigenspace corresponding to each listed eigenvalue. -1- 8 -1 A = 2 3-4 4 6 4,λ = 7 -1
Define T : R3 → R3 by T(p) = p(0) + p(2)t - p(0)t2 - p(2)t3. a. Find T(p) when p(t) = 1 - t2. Is p an eigenvector of T? If p is an eigenvector, what is its eigenvalue? b. Find T(p)
Justify each answer. (T/F) There exists a 2 x 2 matrix that has no eigenvectors in R2.
For the matrices list the eigenvalues, repeated according to their multiplicities. 7 -5 0 0 0 5-3 0 0 0 53 3 0 7 -5 37
P is an n × n stochastic matrix. Justify each answer.(T/F) Every eigenvector of P is a steady state vector.
Find an invertible matrix P and a matrix a C of the form such that the given matrix has the b a form A = PCP-1. D b -b a
Find a basis for the eigenspace corresponding to each listed eigenvalue. Α 3 -1 6 -1 3 6 3 3,λ = -4 2
Define T : R2 → R2 by T (p) = p(1) + p(1)t + p(1)t2. a. Find T(p) when p(t) = 1 + t + t2. Is p an eigenvector of T? If p is an eigenvector, what is its eigenvalue? b. Find T(p) when
Classify the origin as an attractor, repeller, or saddle point of the dynamical system xk+1 = Axk. Find the directions of greatest attraction and/or repulsion. A = 1.7 -.4 .6 .7
Justify each answer. (T/F) If a 5 x 5 matrix A has fewer than 5 distinct eigenvalues, then A is not diagonalizable.
Construct the general solution of x' = Ax involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories. - [3 -9 A = -3 2 23
P is an n × n stochastic matrix. Justify each answer.(T/F) The steady state vector is an eigenvector of P.
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. 402 2 3 4 003
Find an invertible matrix P and a matrix a C of the form such that the given matrix has the b a form A = PCP-1. D b -b a
Construct the general solution of x' = Ax involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories. A = 4 -3 6-2
Find the characteristic polynomial of each matrix using expansion across a row or down a column. 3 0 6 -2 -1 7 3 0 -4
Find a basis for the eigenspace corresponding to each listed eigenvalue. A = 4 -2 -2 0 1 1 0 0,λ = 1,2,3 1
Classify the origin as an attractor, repeller, or saddle point of the dynamical system xk+1 = Axk. Find the directions of greatest attraction and/or repulsion. = [₁ A = .8 -.4 .3 1.5
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. 2 1 -1 2 3 -2 -1 -1 2
Refer to Exercise 4. In the long run, how likely is it for the weather in Edinburgh to be good on a given day?Data from in Exercise 4The weather in Edinburgh is either good, indifferent, or bad
Justify each answer. (T/F) The matrices A and AT have the same eigenvalues, counting multiplicities.
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. 4 1 0 0 4 0 0 0 5
For the matrices list the eigenvalues, repeated according to their multiplicities. 3 -5 3 0 -4 0 0 1 8 0 0 0 0 0 2 1 9-2 -7 19 0 0 0 0 3
Find an invertible matrix P and a matrix a C of the form such that the given matrix has the b a form A = PCP-1. D b -b a
Justify each answer. (T/F) If A is diagonalizable, then the columns of A are linearly independent.
P is an n × n stochastic matrix. Justify each answer.(T/F) The all ones vector is an eigenvector of PT.
Find an invertible matrix P and a matrix a C of the form such that the given matrix has the b a form A = PCP-1. D b -b a
Justify each answer. (T/F) A (square) matrix A is invertible if and only if there is a coordinate system in which the transformation x ↦ Ax is represented by a diagonal matrix.
P is an n × n stochastic matrix. Justify each answer.(T/F) The number 1/2 can be an eigenvalue of a stochastic matrix.
Construct the general solution of x' = Ax involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories. A = 53 -30 90 20 -10 -30 -2 -52 -3 2
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. One eigenvalue is λ = 5 and
It can be shown that the algebraic multiplicity of an eigenvalue λ is always greater than or equal to the dimension of the eigenspace corresponding to λ. Find h in the matrix A below such that the
Find an invertible matrix P and a matrix a C of the form such that the given matrix has the b a form A = PCP-1. D b -b a
Justify each answer. (T/F) A nonzero vector cannot correspond to two different eigenvalues of A.
Construct the general solution of x' = Ax involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories. A = 64 30 -11-23 6 15 23 -9 4
P is an n × n stochastic matrix. Justify each answer.(T/F) The number 2 can be an eigenvalue of a stochastic matrix.
Let A be an n × n matrix, and suppose A has n real eigenvalues, λ1,...,λn, repeated according to multiplicities, so that det(A - λI) = (λ1 - λ)(λ2 - λ)... (λn - λ) Explain why det
Mark each statement. Justify each answer.(T/F) Only linear transformations on finite vectors spaces have eigenvectors.
A and B are n × n matrices. Justify each answer.(T/F) If 0 is an eigenvalue of A, then A is invertible.
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. 5-3 0 3 0 0 0 0 0 9 1-2 2 0 0 2
Find a factorization of the given matrix A in the form A = PCP-1, where C is a block-diagonal matrix with 2 x 2 blocks of the form shown in Example 6. (For each conjugate pair of eigenvalues, use the
Find the B-matrix for the transformation x → Ax when B = {b1, b2, b3}. A = b₂ = -7 1 -3 -2 -48 14 -45 6,b₁ = ]=[] 1 -3 1,b3 , b3 = -3 -16 -19 3 -[J] 0

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