Linear Algebra and Its Applications (5th Edition)
5th Edition
ISBN: 9780321982384
Author: David C. Lay, Steven R. Lay, Judi J. McDonald
Publisher: PEARSON
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Textbook Question
Chapter 5.5, Problem 3E
Let each matrix in Exercises 1–6 act on ℂ2. Find the eigenvalues and a basis for each eigenspace in ℂ2.
3.
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Assume {u1, U2, u3, u4} does not span R³.
Select the best statement.
A. {u1, U2, u3} spans R³ if u̸4 is a linear combination of other vectors in the set.
B. We do not have sufficient information to determine whether {u₁, u2, u3} spans R³.
C. {U1, U2, u3} spans R³ if u̸4 is a scalar multiple of another vector in the set.
D. {u1, U2, u3} cannot span R³.
E. {U1, U2, u3} spans R³ if u̸4 is the zero vector.
F. none of the above
Select the best statement.
A. If a set of vectors includes the zero vector 0, then the set of vectors can span R^ as long as the other vectors
are distinct.
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B. If a set of vectors includes the zero vector 0, then the set of vectors spans R precisely when the set with 0
excluded spans Rª.
○ C. If a set of vectors includes the zero vector 0, then the set of vectors can span Rn as long as it contains n
vectors.
○ D. If a set of vectors includes the zero vector 0, then there is no reasonable way to determine if the set of vectors
spans Rn.
E. If a set of vectors includes the zero vector 0, then the set of vectors cannot span Rn.
F. none of the above
Which of the following sets of vectors are linearly independent? (Check the boxes for linearly independent sets.)
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Chapter 5 Solutions
Linear Algebra and Its Applications (5th Edition)
Ch. 5.1 - Is 5 an eigenvalue of A=[631305226]?Ch. 5.1 - If x is an eigenvector of A corresponding to ,...Ch. 5.1 - Suppose that b1 and b2 are eigenvectors...Ch. 5.1 - If A is an n n matrix and is an eigenvalue of A,...Ch. 5.1 - Is = 2 an eigenvalue of [3238]? Why or why not?Ch. 5.1 - Is = 2 an eigenvalue of [7331]? Why or why not?Ch. 5.1 - Is [14] an eigenvalue of [3138]? If so, find the...Ch. 5.1 - Is [1+21] an eigenvector of [2114]?If so, find the...Ch. 5.1 - Is [431] an eigenvalue of [379451244]? If so, find...Ch. 5.1 - Is [121] an eigenvalue of [367337565]? If so, find...
Ch. 5.1 - Is = 4 an eigenvalue of [301231345]? If so, find...Ch. 5.1 - Is = 3 an eigenvalue of [122321011]? If so, find...Ch. 5.1 - In Exercises 9-16, find a basis for the eigenspace...Ch. 5.1 - In Exercises 9-16, find a basis for the eigenspace...Ch. 5.1 - In Exercises 9-16, find a basis for the eigenspace...Ch. 5.1 - In Exercises 9-16, find a basis for the eigenspace...Ch. 5.1 - In Exercises 9-16, find a basis for the eigenspace...Ch. 5.1 - In Exercises 9-16, find a basis for the eigenspace...Ch. 5.1 - In Exercises 9-16, find a basis for the eigenspace...Ch. 5.1 - In Exercises 9-16, find a basis for the eigenspace...Ch. 5.1 - Find the eigenvalues of the matrices in Exercises...Ch. 5.1 - Find the eigenvalues of the matrices in Exercises...Ch. 5.1 - For A=[123123123], find one eigenvalue, with no...Ch. 5.1 - Without calculation, find one eigenvalue and two...Ch. 5.1 - a. If Ax = x for some vector x. then is an...Ch. 5.1 - a. If Ax = x for some scalar . then x is an...Ch. 5.1 - Explain why a 2 2 matrix can have at most two...Ch. 5.1 - Construct an example of a 2 2 matrix with only...Ch. 5.1 - Let be an eigenvalue of an invertible matrix A....Ch. 5.1 - Show that if A2 is the zero matrix, then the only...Ch. 5.1 - Show that is an eigenvalue of A if and only if ...Ch. 5.1 - Use Exercise 27 to complete the proof of Theorem 1...Ch. 5.1 - Consider an n n matrix A with the property that...Ch. 5.1 - Consider an n n matrix A with the property that...Ch. 5.1 - In Exercises 31 and 32, let A be the matrix of the...Ch. 5.1 - T is the transformation on 3 that rotates points...Ch. 5.1 - Let u and v be eigenvectors of a matrix A, with...Ch. 5.1 - Describe how you might try to build a solution of...Ch. 5.1 - Let u and v be the vectors shown in the figure,...Ch. 5.1 - Repeat Exercise 35, assuming u and v are...Ch. 5.2 - Find the characteristic equation and eigenvalues...Ch. 5.2 - Find the characteristic polynomial and the...Ch. 5.2 - Find the characteristic polynomial and the...Ch. 5.2 - Find the characteristic polynomial and the...Ch. 5.2 - Find the characteristic polynomial and the...Ch. 5.2 - Find the characteristic polynomial and the...Ch. 5.2 - Find the characteristic polynomial and the...Ch. 5.2 - Find the characteristic polynomial and the...Ch. 5.2 - Find the characteristic polynomial and the...Ch. 5.2 - Exercises 914 require techniques from Section 3.1....Ch. 5.2 - Exercises 914 require techniques from Section 3.1....Ch. 5.2 - Exercises 914 require techniques from Section 3.1....Ch. 5.2 - Exercises 914 require techniques from Section 3.1....Ch. 5.2 - Exercises 914 require techniques from Section 3.1....Ch. 5.2 - Exercises 914 require techniques from Section 3.1....Ch. 5.2 - For the matrices in Exercises 1517, list the...Ch. 5.2 - For the matrices in Exercises 15-17, list the...Ch. 5.2 - For the matrices in Exercises 15-17, list the...Ch. 5.2 - It can be shown that the algebraic multiplicity of...Ch. 5.2 - Let A be an n n matrix, and suppose A has n real...Ch. 5.2 - Use a property of determinants to show that A and...Ch. 5.2 - a. The determinant of A is the product of the...Ch. 5.2 - a. If A is 3 3, with columns a1, a2, and a3, then...Ch. 5.2 - A widely used method for estimating eigenvalues of...Ch. 5.2 - Show that if A and B are similar, then det A = det...Ch. 5.2 - Let A = [.6.3.4.7], v1 = [3/74/7], x0 = [.5.5]....Ch. 5.2 - Let A = [abcd]. Use formula (1) for a determinant...Ch. 5.2 - Let A = [.5.2.3.3.8.3.20.4], v1 = [.3.6.1], v2 =...Ch. 5.3 - Compute A8, where A = [4321].Ch. 5.3 - Let A = [31227], v1 = [31], and v2 = [21]. Suppose...Ch. 5.3 - Let A be a 4 4 matrix with eigenvalues 5, 3, and...Ch. 5.3 - In Exercises 1 and 2, let A = PDP1 and compute A4....Ch. 5.3 - In Exercises 1 and 2, let A = PDP1 and compute A4....Ch. 5.3 - In Exercises 3 and 4, use the factorization A =...Ch. 5.3 - In Exercises 3 and 4, use the factorization A =...Ch. 5.3 - In Exercises 5 and 6. the matrix A is factored in...Ch. 5.3 - In Exercises 5 and 6. the matrix A is factored in...Ch. 5.3 - Diagonalize the matrices in Exercises 720, if...Ch. 5.3 - Diagonalize the matrices in Exercises 720, if...Ch. 5.3 - Diagonalize the matrices in Exercises 720, if...Ch. 5.3 - Diagonalize the matrices in Exercises 720, if...Ch. 5.3 - Diagonalize the matrices in Exercises 720, if...Ch. 5.3 - Diagonalize the matrices in Exercises 720, if...Ch. 5.3 - Diagonalize the matrices in Exercises 720, if...Ch. 5.3 - Diagonalize the matrices in Exercises 720, if...Ch. 5.3 - Diagonalize the matrices in Exercises 720, if...Ch. 5.3 - Diagonalize the matrices in Exercises 720, if...Ch. 5.3 - Diagonalize the matrices in Exercises 720, if...Ch. 5.3 - Diagonalize the matrices in Exercises 720, if...Ch. 5.3 - Diagonalize the matrices in Exercises 720, if...Ch. 5.3 - Diagonalize the matrices in Exercises 720, if...Ch. 5.3 - In Exercises 21 and 22, A, B, P, and D are n n...Ch. 5.3 - In Exercises 21 and 22, A, B, P, and D are n n...Ch. 5.3 - A is a 5 5 matrix with two eigenvalues. One...Ch. 5.3 - A is a 3 3 matrix with two eigenvalues. Each...Ch. 5.3 - A is a 4 4 matrix with three eigenvalues. One...Ch. 5.3 - A is a 7 7 matrix with three eigenvalues. One...Ch. 5.3 - Show that if A is both diagonalizable and...Ch. 5.3 - Show that if A has n linearly independent...Ch. 5.3 - A factorization A = PDP1 is not unique....Ch. 5.3 - With A and D as in Example 2, find an invertible...Ch. 5.3 - Construct a nonzero 2 2 matrix that is invertible...Ch. 5.3 - Construct a nondiagonal 2 2 matrix that is...Ch. 5.4 - Find T(a0 + a1t + a1t2), if T is the linear...Ch. 5.4 - Let A, B, and C be n n matrices. The text has...Ch. 5.4 - Let B = b1,b2,b3 and D = d1,d2 be bases for vector...Ch. 5.4 - Let D = {d1, d2} and B = {b1, b2} be bases for...Ch. 5.4 - Let = e1,e2,e3 be the standard basis for 3, B =...Ch. 5.4 - Let B = b1,b2,b3 be a basis for a vector space V...Ch. 5.4 - Let T : 2 3 be the transformation that maps a...Ch. 5.4 - Let T : 2 4 be the transformation that maps a...Ch. 5.4 - Assume the mapping T : 2 2 defined by T(a0 + a1t...Ch. 5.4 - Let B = {b1, b2, b3} be a basis for a vector space...Ch. 5.4 - Define T :2 3 = by T (p) = [p(-1)p(0)p(1)]. a....Ch. 5.4 - Define T : 3 4 by T(p) = [p(-3)p(-1)p(1)p(3)]. a....Ch. 5.4 - In Exercises 11 and 12, find the B-matrix for the...Ch. 5.4 - In Exercises 11 and 12, find the B-matrix for the...Ch. 5.4 - In Exercises 1316, define T : 2 2 by T(x) = Ax....Ch. 5.4 - In Exercises 1316, define T : 2 2 by T(x) = Ax....Ch. 5.4 - In Exercises 1316, define T : 2 2 by T(x) = Ax....Ch. 5.4 - In Exercises 1316, define T : 2 2 by T(x) = Ax....Ch. 5.4 - Let A = [1113] and B = {b1, b2}, for b1 = [11], b2...Ch. 5.4 - Define T : 3 3 by T (x) = Ax, where A is a 3 3...Ch. 5.4 - Verify the statements in Exercises 1924. The...Ch. 5.4 - Verify the statements in Exercises 1924. The...Ch. 5.4 - Verify the statements in Exercises 1924. The...Ch. 5.4 - Verify the statements in Exercises 1924. The...Ch. 5.4 - Verify the statements in Exercises 1924. The...Ch. 5.4 - Verify the statements in Exercises 1924. The...Ch. 5.4 - The trace of a square matrix A is the sum of the...Ch. 5.4 - It can be shown that the trace of a matrix A...Ch. 5.4 - Let V be n with a basis B = {b1 ,, bn}; let W be n...Ch. 5.4 - Let V be a vector space with a basis B = {b1, ,...Ch. 5.4 - Let V be a vector space with a basis B = {b1, ...Ch. 5.4 - [M] In Exercises 30 and 31, find the B-matrix for...Ch. 5.4 - [M] In Exercises 30 and 31, find the B-matrix for...Ch. 5.5 - Show that if a and b are real, then the...Ch. 5.5 - Let each matrix in Exercises 16 act on 2. Find the...Ch. 5.5 - Let each matrix in Exercises 16 act on 2. Find the...Ch. 5.5 - Let each matrix in Exercises 16 act on 2. Find the...Ch. 5.5 - Let each matrix in Exercises 16 act on 2. Find the...Ch. 5.5 - Let each matrix in Exercises 16 act on 2. Find the...Ch. 5.5 - Let each matrix in Exercises 16 act on 2. Find the...Ch. 5.5 - In Exercises 712, use Example 6 to list the...Ch. 5.5 - In Exercises 712, use Example 6 to list the...Ch. 5.5 - In Exercises 712, use Example 6 to list the...Ch. 5.5 - In Exercises 712, use Example 6 to list the...Ch. 5.5 - In Exercises 712, use Example 6 to list the...Ch. 5.5 - In Exercises 712, use Example 6 to list the...Ch. 5.5 - In Exercises 1320, find an invertible matrix P and...Ch. 5.5 - In Exercises 1320, find an invertible matrix P and...Ch. 5.5 - In Exercises 1320, find an invertible matrix P and...Ch. 5.5 - In Exercises 1320, find an invertible matrix P and...Ch. 5.5 - In Exercises 1320, find an invertible matrix P and...Ch. 5.5 - In Exercises 1320, find an invertible matrix P and...Ch. 5.5 - In Exercises 1320, find an invertible matrix P and...Ch. 5.5 - In Exercises 1320, find an invertible matrix P and...Ch. 5.5 - In Example 2, solve the first equation in (2) for...Ch. 5.5 - Let A be a complex (or real) n n matrix, and let...Ch. 5.5 - Chapter 7 will focus on matrices A with the...Ch. 5.5 - Let A be an n n real matrix with the property...Ch. 5.5 - Let A be a real n n matrix, and let x be a vector...Ch. 5.5 - Let A be a real 2 2 matrix with a complex...Ch. 5.6 - The matrix A below has eigenvalues 1, 23, and 13,...Ch. 5.6 - What happens to the sequence {xk } in Practice...Ch. 5.6 - Let A be a 2 2 matrix with eigenvalues 3 and 1/3...Ch. 5.6 - Suppose the eigenvalues of a 3 3 matrix A are 3,...Ch. 5.6 - In Exercises 36, assume that any initial vector x0...Ch. 5.6 - Determine the evolution of the dynamical system in...Ch. 5.6 - In old-growth forests of Douglas fir, the spotted...Ch. 5.6 - Show that if the predation parameter p in Exercise...Ch. 5.6 - Let A have the properties described in Exercise 1....Ch. 5.6 - Prob. 8ECh. 5.6 - In Exercises 914, classify the origin as an...Ch. 5.6 - In Exercises 914, classify the origin as an...Ch. 5.6 - In Exercises 914, classify the origin as an...Ch. 5.6 - In Exercises 914, classify the origin as an...Ch. 5.6 - In Exercises 914, classify the origin as an...Ch. 5.6 - In Exercises 914, classify the origin as an...Ch. 5.6 - Let A = [.40.2.3.8.3.3.2.5]. The vector v1 = [163]...Ch. 5.7 - A real 3 3 matrix A has eigenvalues .5, .2 + .3i,...Ch. 5.7 - A real 3 3 matrix A has eigenvalues .5, .2 + .3i....Ch. 5.7 - A real 3 3 matrix A has eigenvalues 5, .2 + .3i,...Ch. 5.7 - A panicle moving in a planar force field has a...Ch. 5.7 - Let A be a 2 2 matrix with eigenvalues 3 and 1...Ch. 5.7 - In Exercises 36, solve the initial value problem...Ch. 5.7 - In Exercises 36, solve the initial value problem...Ch. 5.7 - In Exercises 36, solve the initial value problem...Ch. 5.7 - In Exercises 36, solve the initial value problem...Ch. 5.7 - In Exercises 7 and 8, make a change of variable...Ch. 5.7 - In Exercises 7 and 8, make a change of variable...Ch. 5.7 - In Exercises 918, construct the general solution...Ch. 5.7 - In Exercises 918, construct the general solution...Ch. 5.7 - In Exercises 918, construct the general solution...Ch. 5.7 - In Exercises 918, construct the general solution...Ch. 5.7 - In Exercises 918, construct the general solution...Ch. 5.7 - In Exercises 918, construct the general solution...Ch. 5.7 - [M] Find formulas for the voltages v1 and v2 (as...Ch. 5.7 - [M] Find formulas for the voltages v1 and v2 for...Ch. 5.7 - [M] Find formulas for the current it and the...Ch. 5.7 - [M] The circuit in the figure is described by the...Ch. 5.8 - How can you tell if a given vector x is a good...Ch. 5.8 - In Exercises 14, the matrix A is followed by a...Ch. 5.8 - In Exercises 14, the matrix A is followed by a...Ch. 5.8 - In Exercises 14, the matrix A is followed by a...Ch. 5.8 - In Exercises 14, the matrix A is followed by a...Ch. 5.8 - Let A = [15162021]. The vectors x, , A5x are...Ch. 5.8 - Let A = [2367]. Repeat Exercise 5, using the...Ch. 5.8 - Exercises 13 and 14 apply to a 3 3 matrix A whose...Ch. 5.8 - Exercises 13 and 14 apply to a 3 3 matrix A whose...Ch. 5.8 - Suppose Ax = x with x 0. Let or be a scalar...Ch. 5.8 - Suppose n is an eigenvalue of the B in Exercise...Ch. 5.8 - A common misconception is that if A has a strictly...Ch. 5 - Mark each statement as True or False. Justify each...Ch. 5 - Show that if x is an eigenvector of the matrix...Ch. 5 - Suppose x is an eigenvector of A corresponding to...Ch. 5 - Use mathematical induction to show that if is an...Ch. 5 - If p(t) = c0 + c1t + c2t2 + + cntn, define p(A)...Ch. 5 - Suppose A = PDP1, where P is 2 2 and D = [2007]....Ch. 5 - Suppose A is diagonalizable and p(t) is the...Ch. 5 - a. Let A be a diagonalizable n n matrix. Show...Ch. 5 - Show that I A is invertible when all the...Ch. 5 - Show that if A is diagonalizable, with all...Ch. 5 - Let u be an eigenvector of A corresponding to an...Ch. 5 - Let G = [AX0B] Use formula (1) for the determinant...Ch. 5 - Use Exercise 12 to find the eigenvalues of the...Ch. 5 - Use Exercise 12 to find the eigenvalues of the...Ch. 5 - Let J be the n n matrix of all 1s, and consider A...Ch. 5 - Apply the result of Exercise 15 to find the...Ch. 5 - Let A = [a11a12a21a22]. Recall from Exercise 25 in...Ch. 5 - Let A = [.4.3.41.2]. Explain why Ak approaches...Ch. 5 - Exercises 1923 concern the polynomial p(t) = a0 +...Ch. 5 - Exercises 1923 concern the polynomial p(t) = a0 +...Ch. 5 - Use mathematical induction to prove that for n 2,...Ch. 5 - Exercises 1923 concern the polynomial p(t) = a0 +...Ch. 5 - Exercises 1923 concern the polynomial p(t) = a0 +...
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