EBK LINEAR ALGEBRA AND ITS APPLICATIONS
6th Edition
ISBN: 9780135851043
Author: Lay
Publisher: PEARSON CO
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Textbook Question
Chapter 4.3, Problem 42E
Exercises 31 and 32 reveal an important connection between linear independence and linear transformations and provide practice using the definition of linear dependence. Let V and W be vector spaces, let T : V → W be a linear transformation, and let {v1,...,vp} be a subset of V.
32. Suppose that T is a one-to-one transformation, so that an equation T(u) = T(v) always implies u = v. Show that if the set of images {T(v1),…,T(vp)} is linearly dependent, then {v1,..., vp} is linearly dependent. This fact shows that a one-to-one linear transformation maps a linearly independent set onto a linearly independent set (because in this case the set of images cannot be linearly dependent).
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Chapter 4 Solutions
EBK LINEAR ALGEBRA AND ITS APPLICATIONS
Ch. 4.1 - Show that the set H of all points in 2 of the form...Ch. 4.1 - Let W = Span{v1,...,vp}, where v1,..,vp are in a...Ch. 4.1 - An n n matrix A is said to be symmetric if AT =...Ch. 4.1 - Let V be the first quadrant in the xy-plane; that...Ch. 4.1 - Let W be the union of the first and third...Ch. 4.1 - Let H be the set of points inside and on the unit...Ch. 4.1 - Construct a geometric figure that illustrates why...Ch. 4.1 - In Exercises 58, determine if the given set is a...Ch. 4.1 - In Exercises 58, determine if the given set is a...Ch. 4.1 - In Exercises 58, determine if the given set is a...
Ch. 4.1 - In Exercises 58, determine if the given set is a...Ch. 4.1 - Let H be the set of all vectors of the form...Ch. 4.1 - Let H be the set of all vectors of the form...Ch. 4.1 - Let W be the set of all vectors of the form...Ch. 4.1 - Let W be the set of all vectors of the form...Ch. 4.1 - Let v1 = [101], v2 = [213], v3 = [426], and w=...Ch. 4.1 - Let v1, v2, v3 be as in Exercise 13, and let w =...Ch. 4.1 - In Exercises 1518, let W be the set of all vectors...Ch. 4.1 - In Exercises 1518, let W be the set of all vectors...Ch. 4.1 - In Exercises 1518, let W be the set of all vectors...Ch. 4.1 - In Exercises 1518, let W be the set of all vectors...Ch. 4.1 - If a mass m is placed at the end of a spring, and...Ch. 4.1 - The set of all continuous real-valued functions...Ch. 4.1 - Determine if the set H of all matrices of the form...Ch. 4.1 - Let F be a fixed 32 matrix, and let H be the set...Ch. 4.1 - Prob. 24ECh. 4.1 - In Exercises 23—32, mark each statement True or...Ch. 4.1 - In Exercises 23—32, mark each statement True or...Ch. 4.1 - In Exercises 23—32, mark each statement True or...Ch. 4.1 - In Exercises 23—32, mark each statement True or...Ch. 4.1 - In Exercises 23—32, mark each statement True or...Ch. 4.1 - Prob. 32ECh. 4.1 - Exercises 2529 show how the axioms for a vector...Ch. 4.1 - Exercises 2529 show how the axioms for a vector...Ch. 4.1 - Exercises 2529 show how the axioms for a vector...Ch. 4.1 - Exercises 2529 show how the axioms for a vector...Ch. 4.1 - Suppose cu = 0 for some nonzero scalar c. Show...Ch. 4.1 - Let u and v be vectors in a vector space V, and...Ch. 4.1 - Let H and K be sub spaces of a vector space V. The...Ch. 4.1 - Given subspaces H and K of a vector space V, the...Ch. 4.1 - Suppose u1,..., up and v1,..., vq are vectors in a...Ch. 4.1 - [M] Show that w is in the subspace of 4 spanned by...Ch. 4.1 - [M] Determine if y is in the subspace of 4 spanned...Ch. 4.1 - [M] The vector space H = Span {1, cos2t, cos4t,...Ch. 4.2 - Let W = {[abc]:a3bc=0}. Show in two different ways...Ch. 4.2 - Let A = [735415524], v = [211], and w = [763]....Ch. 4.2 - Let A be an n n matrix. If Col A = Nul A, show...Ch. 4.2 - Determine if w = [134] is in Nul A, where A =...Ch. 4.2 - Determine if w = [532] is in Nul A, where A =...Ch. 4.2 - In Exercises 36, find an explicit description of...Ch. 4.2 - In Exercises 36, find an explicit description of...Ch. 4.2 - In Exercises 36, find an explicit description of...Ch. 4.2 - In Exercises 36, find an explicit description of...Ch. 4.2 - In Exercises 714, either use an appropriate...Ch. 4.2 - In Exercises 714, either use an appropriate...Ch. 4.2 - In Exercises 714, either use an appropriate...Ch. 4.2 - In Exercises 714, either use an appropriate...Ch. 4.2 - In Exercises 714, either use an appropriate...Ch. 4.2 - In Exercises 714, either use an appropriate...Ch. 4.2 - In Exercises 714, either use an appropriate...Ch. 4.2 - In Exercises 714, either use an appropriate...Ch. 4.2 - In Exercises 15 and 16, find A such that the given...Ch. 4.2 - Prob. 16ECh. 4.2 - For the matrices in Exercises 1720, (a) find k...Ch. 4.2 - For the matrices in Exercises 1720, (a) find k...Ch. 4.2 - For the matrices in Exercises 1720, (a) find k...Ch. 4.2 - For the matrices in Exercises 17-20, (a) find k...Ch. 4.2 - With A as in Exercise 17, find a nonzero vector in...Ch. 4.2 - With A as in Exercise 3, find a nonzero vector in...Ch. 4.2 - Let A=[61236] and w=[21]. Determine if w is in Col...Ch. 4.2 - Let A=[829648404] and w=[212]. Determine w is in...Ch. 4.2 - In Exercises 25—38, A denotes an mn matrix. Mark...Ch. 4.2 - Prob. 26ECh. 4.2 - Prob. 27ECh. 4.2 - Prob. 28ECh. 4.2 - Prob. 29ECh. 4.2 - Prob. 30ECh. 4.2 - In Exercises 25—38, A denotes an mn matrix. Mark...Ch. 4.2 - Prob. 35ECh. 4.2 - Prob. 37ECh. 4.2 - Prob. 38ECh. 4.2 - It can be shown that a solution of the system...Ch. 4.2 - Consider the following two systems of equations:...Ch. 4.2 - Prove Theorem 3 as follows: Given an m n matrix...Ch. 4.2 - Let T : V W be a linear transformation from a...Ch. 4.2 - Define T : p2 by T(p)=[p(0)p(1)]. For instance, if...Ch. 4.2 - Define a linear transformation T: p2 2 by...Ch. 4.2 - Let M22 be the vector space of all 2 2 matrices,...Ch. 4.2 - (Calculus required) Define T : C[0, 1 ] C[0, 1]...Ch. 4.2 - Let V and W be vector spaces, and let T : V W be...Ch. 4.2 - [M] Determine whether w is in the column space of...Ch. 4.2 - [M] Determine whether w is in the column space of...Ch. 4.2 - [M] Let a1,,a5 denote the columns of the matrix A,...Ch. 4.2 - [M] Let H = Span {v1, v2} and K = Span {v3, v4},...Ch. 4.3 - Let v1=[123] and v2=[279]. Determine if {v1, v2}...Ch. 4.3 - Let v1=[134], v2=[621], v3=[223], and v4=[489]....Ch. 4.3 - Let v1=[100], v2=[010], and H={[ss0]:sin}. Then...Ch. 4.3 - Let V and W be vector spaces, let T : V W and U :...Ch. 4.3 - Determine which sets in Exercises 1-8 are bases...Ch. 4.3 - Determine which sets in Exercises 1-8 are bases...Ch. 4.3 - Determine which sets in Exercises 1-8 are bases...Ch. 4.3 - Determine which sets in Exercises 1-8 are bases...Ch. 4.3 - Determine which sets in Exercises 1-8 are bases...Ch. 4.3 - Determine which sets in Exercises 1-8 are bases...Ch. 4.3 - Determine which sets in Exercises 1-8 are bases...Ch. 4.3 - Determine which sets in Exercises 1-8 are bases...Ch. 4.3 - Find bases for the null spaces of the matrices...Ch. 4.3 - Find bases for the null spaces of the matrices...Ch. 4.3 - Find a basis for the set of vectors in 3 in the...Ch. 4.3 - Find a basis for the set of vectors in 2 on the...Ch. 4.3 - In Exercises 13 and 14, assume that A is row...Ch. 4.3 - In Exercises 13 and 14, assume that A is row...Ch. 4.3 - In Exercises 15-18, find a basis for the space...Ch. 4.3 - In Exercises 15-18, find a basis for the space...Ch. 4.3 - In Exercises 15-18, find a basis for the space...Ch. 4.3 - In Exercises 15-18, find a basis for the space...Ch. 4.3 - Let v1=[437], v2=[192], v3=[7116], and H =...Ch. 4.3 - Let v1=[7495], v2=[4725], v3=[1534]. It can be...Ch. 4.3 - Prob. 21ECh. 4.3 - Prob. 22ECh. 4.3 - Prob. 23ECh. 4.3 - Prob. 24ECh. 4.3 - Prob. 25ECh. 4.3 - Prob. 26ECh. 4.3 - Prob. 27ECh. 4.3 - Suppose 4 = Span {v1,,v4}. Explain why {v1,,v4} is...Ch. 4.3 - Let B = {v1,..., vn} be a linearly independent set...Ch. 4.3 - Let v1=[101], v2=[011], v3=[010], and let H be the...Ch. 4.3 - In the vector space of all real-valued functions,...Ch. 4.3 - Let V be the vector space of functions that...Ch. 4.3 - (RLC circuit) The circuit in the figure consists...Ch. 4.3 - Exercises 29 and 30 show that every basis for n...Ch. 4.3 - Exercises 29 and 30 show that every basis for n...Ch. 4.3 - Exercises 31 and 32 reveal an important connection...Ch. 4.3 - Exercises 31 and 32 reveal an important connection...Ch. 4.3 - Consider the polynomials p1(t) = 1 + t2 and p2(t)...Ch. 4.3 - Consider the polynomials p1(t) = 1 + t, p2(t) = 1 ...Ch. 4.3 - Let V be a vector space that contains a linearly...Ch. 4.3 - [M] Let H = Span {u1, u2, u3} and K = Span{v1,v2,...Ch. 4.3 - [M] Show that {t, sin t, cos 2t, sin t cos t} is a...Ch. 4.4 - Let b1=[100], b2=[340], b3=[363], and x=[823]. a....Ch. 4.4 - The set B = {1 + t, 1 + t2, t + t2} is a basis for...Ch. 4.4 - In Exercises 1-4, find the vector x determined by...Ch. 4.4 - In Exercises 1-4, find the vector x determined by...Ch. 4.4 - In Exercises 1-4, find the vector x determined by...Ch. 4.4 - In Exercises 1-4, find the vector x determined by...Ch. 4.4 - In Exercises 5-8, find the coordinate vector [ x...Ch. 4.4 - In Exercises 5-8, find the coordinate vector [ x...Ch. 4.4 - In Exercises 5-8, find the coordinate vector [ x...Ch. 4.4 - In Exercises 5-8, find the coordinate vector [ x...Ch. 4.4 - In Exercises 9 and 10, find the...Ch. 4.4 - In Exercises 9 and 10, find the...Ch. 4.4 - In Exercises 11 and 12, use an inverse matrix to...Ch. 4.4 - In Exercises 11 and 12, use an inverse matrix to...Ch. 4.4 - The set B = {1 + t2, t + t2, 1 + 2t + t2} is a...Ch. 4.4 - The set B = {1 t2, t t2, 2 2t + t2} is a basis...Ch. 4.4 - Prob. 15ECh. 4.4 - The vectors v1=[13], v2=[28], v3=[37] span 2 but...Ch. 4.4 - Let B = {b1,...,bn} be a basis for a vector space...Ch. 4.4 - Let S be a finite set in a vector space V with the...Ch. 4.4 - Suppose {v1,...,v4} is a linearly dependent...Ch. 4.4 - Let B={[14],[29]}. Since the coordinate mapping...Ch. 4.4 - Exercises 23-26 concern a vector space V, a basis...Ch. 4.4 - Exercises 23-26 concern a vector space V, a basis...Ch. 4.4 - Exercises 23-26 concern a vector space V, a basis...Ch. 4.4 - Exercises 23-26 concern a vector space V, a basis...Ch. 4.4 - In Exercises 27-30, use coordinate vectors to test...Ch. 4.4 - In Exercises 27-30, use coordinate vectors to test...Ch. 4.4 - In Exercises 27-30, use coordinate vectors to test...Ch. 4.4 - In Exercises 27-30, use coordinate vectors to test...Ch. 4.4 - Use coordinate vectors to test whether the...Ch. 4.4 - Let p1 (t) = 1 + t2, p2(t) = t 3t2, p3 (t) = 1 +...Ch. 4.4 - In Exercises 33 and 34, determine whether the sets...Ch. 4.4 - In Exercises 33 and 34, determine whether the sets...Ch. 4.4 - Prob. 39ECh. 4.4 - [M] Let H = Span{v1,v2, v3} and B ={v1,v2, v3}....Ch. 4.4 - [M] Exercises 37 and 38 concern the crystal...Ch. 4.4 - [M] Exercises 37 and 38 concern the crystal...Ch. 4.5 - Decide whether each statement is True or False,...Ch. 4.5 - Let H and K be subspaces of a vector space V. In...Ch. 4.5 - For each subspace in Exercises 1-8, (a) find a...Ch. 4.5 - For each subspace in Exercises 1-8, (a) find a...Ch. 4.5 - For each subspace in Exercises 1-8, (a) find a...Ch. 4.5 - For each subspace in Exercises 1-8, (a) find a...Ch. 4.5 - For each subspace in Exercises 1-8, (a) find a...Ch. 4.5 - For each subspace in Exercises 1-8, (a) find a...Ch. 4.5 - For each subspace in Exercises 1-8, (a) find a...Ch. 4.5 - For each subspace in Exercises 1-8, (a) find a...Ch. 4.5 - In Exercises 11 and 12, find the dimension of the...Ch. 4.5 - In Exercises 11 and 12, find the dimension of the...Ch. 4.5 - Determine the dimensions of Nul A and Col A for...Ch. 4.5 - Determine the dimensions of Nul A and Col A for...Ch. 4.5 - Determine the dimensions of Nul A and Col A for...Ch. 4.5 - Determine the dimensions of Nul A and Col A for...Ch. 4.5 - Determine the dimensions of Nul A, Col A, and Row...Ch. 4.5 - Determine the dimensions of Nul A and Col A for...Ch. 4.5 - In Exercises 17—26, V is a vector space and A is...Ch. 4.5 - Prob. 18ECh. 4.5 - Prob. 22ECh. 4.5 - Prob. 24ECh. 4.5 - Prob. 25ECh. 4.5 - Prob. 26ECh. 4.5 - The first four Hermite polynomials are 1, 2t, 2 +...Ch. 4.5 - The first four Laguerre polynomials are 1, 1 t, 2...Ch. 4.5 - Let B be the basis of 3 consisting of the Hermite...Ch. 4.5 - Let S be a subset of an n-dimensional vector space...Ch. 4.5 - Let H be an n-dimensional subspace of an...Ch. 4.5 - Prob. 36ECh. 4.6 - Let B = {b1, b2} and C = {c1, c2} be bases for a...Ch. 4.6 - Let B = {b1, b2} and C = {c1, c2} be bases for a...Ch. 4.6 - Let u = {u1, u2} and w = {w1, w2} be bases for V,...Ch. 4.6 - Let A = {a1, a2, a3} and D = {d1, d2, d3} be bases...Ch. 4.6 - Let A = {a1, a2, a3} and B = {b1, b2, b3} be bases...Ch. 4.6 - Let D = {d1, d2, d3} and F = {f1, f2, f3} be bases...Ch. 4.6 - In Exercises 7-10, let B = {b1, b2} and C = {c1,...Ch. 4.6 - In Exercises 7-10, let B = {b1, b2} and C = {c1,...Ch. 4.6 - In Exercises 7-10, let B = {b1, b2} and C = {c1,...Ch. 4.6 - In Exercises 7-10, let B = {b1, b2} and C = {c1,...Ch. 4.6 - In 2 find the change-of-coordinates matrix from...Ch. 4.6 - In 2 find the change-of-coordinates matrix from...Ch. 4.6 - Exercises 15 and 16 provide a proof of Theorem 15....Ch. 4.6 - Prob. 18ECh. 4.6 - Prob. 19ECh. 4.6 - [M] Let P=[121350461],v1=[223],v2=[852],v3=[726]...Ch. 4.6 - Let B = {b1, b2}, C = {c1, c2}, and D = {d1, d2}...Ch. 4.8 - Verify that the signals in Exercises 1 and 2 are...Ch. 4.8 - Prob. 2ECh. 4.8 - Prob. 3ECh. 4.8 - Show that the signals in Exercises 3-6 form a...Ch. 4.8 - Show that the signals in Exercises 3-6 form a...Ch. 4.8 - Show that the signals in Exercises 3-6 form a...Ch. 4.8 - Prob. 7ECh. 4.8 - Prob. 8ECh. 4.8 - Prob. 9ECh. 4.8 - Prob. 10ECh. 4.8 - Prob. 11ECh. 4.8 - Prob. 12ECh. 4.8 - In Exercises 13-16, find a basis for the solution...Ch. 4.8 - In Exercises 13-16, find a basis for the solution...Ch. 4.8 - In Exercises 13-16, find a basis for the solution...Ch. 4.8 - In Exercises 13-16, find a basis for the solution...Ch. 4.8 - Exercises 17 and 18 concern a simple model of the...Ch. 4.8 - Exercises 17 and 18 concern a simple model of the...Ch. 4.8 - Prob. 21ECh. 4.8 - A lightweight cantilevered beam is supported at N...Ch. 4.8 - Prob. 25ECh. 4.8 - Prob. 26ECh. 4.8 - Prob. 27ECh. 4.8 - Prob. 28ECh. 4.8 - Prob. 29ECh. 4.8 - Prob. 30ECh. 4.8 - Prob. 31ECh. 4.8 - Write the difference equations in Exercises 29 and...Ch. 4.8 - Prob. 33ECh. 4.8 - Prob. 34ECh. 4.8 - Let yk = k2 and zk = 2k|k|. Are the signals {yk}...Ch. 4 - Find a basis for the set of all vectors of the...Ch. 4 - Let u1=[246], u2=[125], b=[b1b2b3], and W =...Ch. 4 - Explain what is wrong with the following...Ch. 4 - Consider the polynomials p1(t) = 1 +t, p2(t) = 1 ...Ch. 4 - Prob. 24SECh. 4 - Prob. 25SECh. 4 - Prob. 26SECh. 4 - Let T : n m be a linear transformation. a. What...Ch. 4 - Prob. 28SECh. 4 - Let S be a finite minimal spanning set of a vector...Ch. 4 - Prob. 30SECh. 4 - Prob. 33SECh. 4 - The concept of rank plays an important role in the...Ch. 4 - Determine if the matrix pairs in Exercises 19-22...Ch. 4 - Determine if the matrix pairs in Exercises 19-22...Ch. 4 - Determine if the matrix pairs in Exercises 19-22...Ch. 4 - Prob. 40SE
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