New Semester Started Get 50% OFF Study Help! --h --m --s Claim Now

Question Answers

Textbooks

Find textbooks, questions and answers

Oops, something went wrong!

Change your search query and then try again

Result Not Found

Books FREE
Study Help
Tutors
AI Study Planner NEW
Sell Books

Search
Sign In
Register

study help
mathematics
linear algebra

Linear Algebra and Its Applications 4th edition David C. Lay - Solutions

Given subspaces H and K of a vector space V, the sum of H and K, written as H + K, is the set of all vectors in V that can be written as the sum of two vectors, one in H and the other in K; that is, H + K = {w: w = u + v for some u in H and some v in K} a. Show that H + K is a subspace of V. b.
Suppose u1, ....., up and v1,....., vq are vectors in a vector space V, and letH = Span {u1,.....,up} and K = Span {v1, ....... , vq}Show that H + K = Span {u1,......,up, v1,......., vq}?
Show that w is in the subspace of R4 spanned by v1, v2, v3 where
Determine if y is in the subspace of R4 spanned by the columns of A, where
The vector space H = Span {1, cos2 t, cos4 t, cos6 t} contains at least two interesting functions that will be used in a later exercise:Study the graph of f for 0 ( t ( 2(, and guess a simple formula for f(t). Verify your conjecture by graphing the dif between 1 C f(t) and your formula for f(t).
The vector space H = Span {1, cos2 t, cos4 t, cos6 t} contains at least two interesting functions that will be used in a later exercise:Study the graph of f for 0 ( t ( 2(, and guess a simple formula for f(t). Verify your conjecture by graphing the dif between 1 C f(t) and your formula for f(t).
Construct a geometric figure that illustrates why a line in R2 not through the origin is not closed under vector addition?
In Exercise 1-2, determine if the given set is a subspace of Pn for an appropriate value of n. Justify your answer? 1. All polynomials of the form p(t) = at2, where a is in R. 2. All polynomials of the form p(t) = a + t2 where a is in R?
Let H be the set of all vectors of the formFind a vector v in R3 such that H = Span {v}. Why does this show that H is a subspace of R3?
In Exercise 1-2, find an explicit description of Nul A, by listing vectors that span the null space.1.2.
In Exercises 1 and 2, find A such that the given set is Col A?1.2.
For the matrices in Exercises 1-2, (a) find k such that Nul A is a subspace of Rk, and (b) find k such that Col A is a subspace of Rk.1.2.
With A as in Exercise 17, find a nonzero vector in Nul A and a nonzero vector in Col A?
With A as in Exercise 18, find a nonzero vector in Nul A and a nonzero vector in Col A?
I can be shown that a solution of the system below is x1 = 3, x2 = 2, and x3 = - 1. Use this fact and the theory from this section to explain why another solution is x1 = 30, x2 = 20, and x3 = - 10, (Observe how the solutions are related, but make no other calculations).
Consider the following two systems of equations:
Prove Theorem 3 as follows: Given an m x n matrix A, an element in Col A has the form Ax for some x in Rn. Let Ax and Aw represent any two vectors in Col A. a. Explain why the zero vector is in Col A. b. Show that the vector Ax C Aw is in Col A. c. Given a scalar c, show that c(Ax) is in Col A.
In Exercise 1 - 2, either use an appropriate theorem to show that the given set, W, is a vector space, or find specific example to the contrary?1. 2.
Is Î» = 2 an eigenvalue ofWhy or why not?
Find the eigenvalues of the matrices in Exercises 17 and 18.1.2.
Is Î» = -3 an eigenvalue ofWhy or why not?
Without calculation, find one eigenvalue and two linearly independent eigenvectors ofJustify your answer
a. If Ax = Ax for some vector x, then A is an eigenvalue of A. b. A matrix A is not invertible if and only if 0 is an eigenvalue of A.
a. If Ax = λx for some scalar A, then x is an eigenvector of A. b. If v1 and v2 are linearly independent eigenvectors, then they correspond to distinct eigenvalues. c. A steady-state vector for a stochastic matrix is actually an eigenvector. d. The eigenvalues of a matrix are on its main
Explain why a 2 x 2 matrix can have at most two distinct eigenvalues. Explain why an n x n matrix can have at most n distinct eigenvalues.
Construct an example of a 2 × 2 matrix with only one distinct eigenvalue.
Let A be an eigenvalue of an invertible matrix A. Show that A-1 is an eigenvalue of λ-1.
Show that if A2 is the zero matrix, then the only eigenvalue of λ is 0.
Show that λ is an eigenvalue of A if and only if λ is an eigenvalue of AT.
Use Exercise 27 to complete the proof of Theorem 1 for the case in which A is lower triangular.
T is the transformation on R2 that reflects points across some line through the origin.
T is the transformation on R3 that rotates points about some line through the origin.
Let u and v be eigenvectors of a matrix A, with corresponding eigenvalues A λ and μ and let c1 and c2 be scalars. Define xk = c1λku + c2μkv (k = 0, 1, 2,...) a. What is xk+1, by definition? b. Compute Axk from the formula for xk, and show that Axk = xk+1. This calculation will prove that the
Describe how you might try to build a solution of a difference equation xk+1 = Axk (k = 0, 1, 2,...) if you were given the initial x0 and this vector did not happen to be an eigenvector of A.
Let u and v be the vectors shown in the figure, and suppose u and v are eigenvectors of a 2 x 2 matrix A that correspond to eigenvalues 2 and 3, respectively. Let T: R2 †’ R2 be the linear transformation given by T(x) = Ax for each x in R2, and let w = u + v. Make a copy of the figure,
In Exercises 1-2, use a matrix program to find the eigen- 40. values of the matrix. Then use the method of Example 4 with a row reduction routine to produce a basis for each eigenspace.1.2.
Is Î» = 4 an eigenvalue ofIf so, find one corresponding eigenvector.
Is Î» = 1 an eigenvalue ofIf so, find one corresponding eigenvector.
In Exercises 9-16, find a basis for the eigenspace corresponding to each listed eigenvalue.1.2.
Find the characteristic polynomial and the real eigenvalues of the matrices in Exercises 1-2.1.2.
For the matrices in Exercises 1-2, list the real eigenvalues, repeated according to their multiplicities.1.2.
It can be shown that the algebraic multiplicity of an eigenvalue A is always greater than or equal to the dimension of the eigenspace corresponding to Î». Find h in the matrix A below such that the eigenspace for Î» = 4 is two-dimensional:
Show that if A and B are similar, then det A = det B.
LetUse formula (1) for a determinant (given before example 2) to show that det A = ad - bc. Consider two case: a ‰ 0 and a = 0.
a. Show that v1, v2, v3 are eigenvectors of A.b. Let x0 be any vector in R3 with nonnegative entries whose sum is 1. (In Section 4.9, x0 was called a probability vector.) Explain why there are constants c1, c2, c3 such that x0 = c1v1 = c2v2 + c3v3. Compute wr x0, and deduce that c1 = 1.c. For k =
For each value of a in the set {32, 31.9, 31.8, 32.1,32.2}, compute the characteristic polynomial of A and the eigenvalues. In each case, create a graph of the characteristic polynomial p(t) = det (A - fl) for 0
Exercises 9-14 require techniques from Section 3.1. Find the characteristic polynomial of each matrix, using either a cofactor expansion or the special formula for 3 x 3 determinants described prior to Exercises 15-18 in Section 3.1.1.2.
In exercise 1 and 2, let A = PDP-1 and compute A4,1.2.
In Exercises 1 and 2, A, B, P, and D are n x n matrices. Mark each statement True or False. Justify each answer. 1. a. A is diagonalizable if A = PDP-l for some matrix D and some invertible matrix P. b. If Rn has a basis of eigenvectors of A, then A is diagonalizable. c. A is diagonalizable if and
A is a 5 x 5 matrix with two eigenvalues. One eigenspace is three-dimensional, and the other eigenspace is two-dimensional. Is A diagonalizable? Why?
A is a 3 x 3 matrix with two eigenvalues. Each eigenspace is one-dimensional. Is A diagonalizable? Why?
A is a 4 x 4 matrix with three eigenvalues. One eigenspace is one-dimensional, and one of the other eigenspaces is two-dimensional. Is it possible that A is not diagonalizable? Justify your answer.
A is a 7 x 7 matrix with three eigenvalues. One eigenspace is two-dimensional, and one of the other eigenspaces is three-dimensional. Is it possible that A is not diagonalizable? Justify your answer.
Show that if A is both diagonalizable and invertible, then so is A-1.
Show that if A has n linearly independent eigenvectors, then so does AT.
A factorization A = PDP-l is not unique. Demonstrate this for the matrix A in Example 2. WithUse the information in Example 2 to find a matrix P1 such that A = P1D1P1-1.
In exercise 1 and 2 the factorization A = PDP-1 to compute Ak, where k represents an arbitrary positive integer.1.2.
Construct a nonzero 2 x 2 matrix that is invertible but not diagonalizable.
Construct a nondiagonal 2 x 2 matrix that is diagonalizable but not invertible.
Diagonalize the matrices in Exercises 1-2. Use your matrix program's eigenvalue command to find the eigenvalues, and then compute bases for the eigenspaces as in Section 5.1.1.2.
In Exercises 1 and 2, the matrix A is factored in the form PDP-l. Use the Diagonalization Theorem to find the eigenvalues of A and a basis for each eigenspace.1.2.
Diagonalize the matrices in Exercises 7-20, if possible. The real eigenvalues for Exercises 1-2 are included below the matrix.1.2.
Let B = {b1, b2, b3} and D = {d1, d2} be bases for vector spaces V and W, respectively. Let T: V †’ W be a linear transformation with the property thatFind the matrix for T relative to B and D.
Define T:a. Show that T is a linear transformation. b. Find the matrix for T relative to the basis {1, t, t2, t3} for P3 and the standard basis for R4.
In Exercises 1 and 2, find the B-matrix for the transformation x †’ Ax, where B = {b1, b2}.1.2.
In Exercises 13-16, define T: R2 †’ R2 by T(x) = Ax. Find a basis B for R2 with the property that [T]B is diagonal.1.2.
Define T: R3 → R3 by T(x) = Ax, where A is a 3 x 3 matrix with eigenvalues 5, 5, and -2. Does there exist a basis B for R3 such that the B-matrix for T is a diagonal matrix? Discuss.
Verify the statements in Exercises 1-2. The matrices are square. 1. If A is similar to B, then A2 is similar to B2. 2. If B is similar to A and C is similar to A, then B is similar to C.
Let D = {d1, d2} and B = {b1, b2} be bases for vector spaces V and W, respectively. Let T: V †’ W be a linear transformation with the property thatFind the matrix for T relative to D and B.
The trace of a square matrix A is the sum of the diagonal entries in A and is denoted by tr A. It can be verified that tr(FG) = tr(GF) for any two n x n matrices F and G. Show that if A and B are similar, then tr A = tr B.
It can be shown that the trace of a matrix A equals the sum of the eigenvalues of A. Verify this statement for the case when A is diagonalizable.
Let V be Rn with a basis B = {b1... bn}; let W be Rn with the standard basis, denoted here by E; and consider the identity transformation I: Rn ! Rn, where 7(x) = x. Find the matrix for I relative to B and E. What was this matrix called in Section 4.4?
Let V be a vector space with a basis B = {b1, ....bn) Let W be the same space V with a basis C = {c1,... c2) be the identity transformation I : V → W. Find the matrix for I relative to B and C. What was this matrix called in Section 4.7?
Let V be a vector space with a basis B = {b1,..., bn}. Find the B-matrix for the identity transformation I : V → V.
Let Îµ = {e1, e2, e3} be the standard basis for R3, let B = {b1, b2, b3} be a basis for a vector space V, and let T: R3 †’ V be a linear transformation with the property thata. Compute T(e1), T(e2), and T(e3). b. Compute [T(e1)]B, [T(e2)B, and [T(e3)B. c. Find the matrix for T
In Exercises 1 and 2, find the B-matrix for the transformation x †’ Ax where B = {b1, b2, b3}.1.2.
Let T be the transformation whose standard matrix is given below. Find a basis for R4 with the property that [T]B is diagonal.
Let B = {b1, b2, b3} be a basis for a vector space V and let T: V †’ R2 be a linear transformation with the property thatFind the matrix for T relative to B and the standard basis for R2.
Let T: P2 → P3 be the transformation that maps a polyno¬mial p(f) into the polynomial (t + 3)p(f). a. Find the image of p(f) = 3 - 2t C t2. b. Show that T is a linear transformation. c. Find the matrix for T relative to the bases {1, t, t2} and {1, t, t2, t3}.
Let T: P2 → P4 be the transformation that maps a polyno¬mial p(t) into the polynomial p(t) C 2t2p(t). a. Find the image of p(t) = 3 - 2t C t2. b. Show that T is a linear transformation. c. Find the matrix for T relative to the bases {1, t, t2} and {1, t, t2, t3, t4}.
Assume the mapping T: P2 †’ P2 defined byIs linear. Find the matrix representation of T relative to the basis B = [1, t, t2].
Let B = {b1, b2, b3} be a basis for a vector space V. Find T(4b1 - 3b2) when T is a linear transformation from V to V whose matrix relative to 13 is
Let each matrix in Exercise 1 act on C2. Find the eigenvalues and a basis for each eigenspaces in C2.
In Exercises 13-20, find an invertible matrix P and a matrix C of the formSuch that the given matrix has the form A = PCP-1
In Example 2, solve the first equation in (2) for x2 in terms of x1, and from that produce the eigenvectorfor the matrix A. Show that this y is a (complex) multiple of the vector v1 used in Example 2.
Let A be an n x n real matrix with the property that AT = A, let x be any vector in Cn, and let q = xrAx. The equalities below show that q is a real number by verifying that q = q. Give a reason for each step.
Let A be an n x n real matrix with the property that AT = A. Show that if Ax = λx for some nonzero vector x in Cn, then, in fact, A is real and the real part of x is an eigenvector of A.
Let A be a real n x n matrix, and let n x n be a vector in Cn. Show that Re(Ax) = A(Re x) and Im(Ax) = A(Im x).
Let A be a real 2 x 2 matrix with a complex eigenvalue λ = a - bi (b ≠ 0) and an associated eigenvector v in C2. a. Show that A(Re v) = a Rev + b Im v and A(Im v) = - b Rev + a Im v. b. Verify that if P and C are given as in Theorem 9, then AP = PC.
In Exercises 27 and 28, 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 real and imaginary parts of one eigenvector in C4 to create two columns of
Let A be a 2 Ã— 2 matrix with eigenvalues 3 and 1/3 and corresponding eigenvectors
Produce the general solution of the dynamical system xk+1 = Axk when A is the stochastic matrix for the Hertz Rent A Car model in Exercise 16 of Section 4.9.
Construct a stage-matrix model for an animal species that has two life stages: juvenile (up to 1 year old) and adult. Suppose the female adults give birth each year to an average of 1.6 female juveniles. Each year, 30% of the juveniles survive to become adults and 80% of the adults survive. For k >
A herd of American buffalo (bison) can be modeled by a stage matrix similar to that for the spotted owls. The females can be divided into calves (up to 1 year old), yearlings (1 to 2 years), and adults. Suppose an average of 42 female calves are born each year per 100 adult females. (Only adults
Suppose the eigenvalues of a 3 Ã— 3 matrix A are 3, 4/5, and 3/5, with corresponding eigenvectorsAnd Let
Determine the evolution of the dynamical system in Example 1 when the predation parameter p is .2 in equation (3). (Give a formula for xk.) Does the owl population grow or decline? What about the wood rat population?
Determine the evolution of the dynamical system in Example 1 when the predation parameter p is .125. (Give a formula for xk.) As time passes, what happens to the sizes of the owl and wood rat populations? The system tends toward what is sometimes called an unstable equilibrium. What do you think
In old-growth forests of Douglas fir, the spotted owl dines mainly on flying squirrels. Suppose the predator-prey matrix for these two populations isthe predation parameter p is .325, both populations grow. Estimate the long-term growth rate and the eventual ratio of owls to flying squirrels.
Show that if the predation parameter p in Exercise 5 is .5, both the owls and the squirrels will eventually perish. Find a value of p for which populations of both owls and squirrels tend toward constant levels. What are the relative population sizes in this case?
Let A have the properties described in Exercise 1. a. Is the origin an attractor, a repeller, or a saddle point of the dynamical system Xk+1 = Axk? b. Find the directions of greatest attraction and/or repulsion for this dynamical system. c. Make a graphical description of the system, showing the

Showing 3200 - 3300 of 11883

First
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
Last

Step by Step Answers

Services

Sitemap
Fun
Definitions
Become Tutor
Used Textbooks
Study Help Categories
Recent Questions
Expert Questions
Campus Wear
Sell Your Books

Company Info

Security
Copyrights
Privacy Policy
Terms & Conditions
SolutionInn Fee
Scholarship
Online Quiz
Give Feedback, Get Rewards

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Student Discount
Campus Ambassador

Secure Payment

Download Our App

© 2026 SolutionInn. All Rights Reserved