Differential Equations and Linear Algebra (4th Edition)
Differential Equations and Linear Algebra (4th Edition)
4th Edition
ISBN: 9780321964670
Author: Stephen W. Goode, Scott A. Annin
Publisher: PEARSON
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Linear Algebra
In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied unde…
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Chapter 4.3, Problem 18P

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For Problems 3 − 22 , express S in set notation and determine whether it is a subspace of the given vector space V .

V is the vector space of all real valued functions defined on the interval ( − ∞ , ∞ ) , and S is the subset of V consisting of all real valued functions satisfying f ( − x ) = f ( x ) for all x ∈ ( − ∞ , ∞ ) .

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(3) For each of the following sets, determine whether it is linearly independent or dependent. (0) € 9000 3 (c) {x³x, 2x² + 4,-2x³ +3x²+2x+6} in P₂ (R), where P3 (R) denotes the vector space of polynomials over the reals of degree at most 3. -1 2 (d) {(-12 1) (1 7¹). (1¹3) (3¹2)} in M2x2(R), where M2x2(R) 0 denotes the vector space of 2 × matrices over the reals. {0·0)} span(S), i.e., orthogonal to every vector in span(S). (b) (4) Let S in R4. 7 in R4. CR4. Find all vectors in R4 that are orthogonal to (5) Let A and B be matrices over the reals. For each of the following statements, determine whether it is true or false. If it is true, prove it. If it is false, give a counter example to disprove it. (a) If A + B is defined, then rank(A + B) = rank(A) + rank(B). (b) If AB is defined, then rank(AB) = rank(A) rank(B). (c) If A has size m × n, then rank(A) ≤ min{m, n}. (d) rank(A) = rank(At) (e) nullity(A) = nullity(A¹)

Chapter 4 Solutions

Differential Equations and Linear Algebra (4th Edition)

Ch. 4.1 - True-False Review For items a-I, decide if the...Ch. 4.1 - Prob. 2TFRCh. 4.1 - For items a-I, decide if the given statement is...Ch. 4.1 - For items a-I, decide if the given statement is...Ch. 4.1 - For items a-I, decide if the given statement is...Ch. 4.1 - For items a-I, decide if the given statement is...Ch. 4.1 - Prob. 7TFRCh. 4.1 - Prob. 8TFRCh. 4.1 - For items a-I, decide if the given statement is...Ch. 4.1 - Prob. 10TFR
Ch. 4.1 - Prob. 11TFRCh. 4.1 - Prob. 12TFRCh. 4.1 - If x=(1,4) and y=(5,1), determine the vectors...Ch. 4.1 - If x=(3,1) and y=(1,2), determine the vectors...Ch. 4.1 - If x=(5,2,9) and y=(1,6,4), determine the additive...Ch. 4.1 - If x=(3,1,2,5) and y=(1,2,9,2), determine...Ch. 4.1 - If x=(1,2,3,4,5) and z=(1,0,4,1,2) and y is 5 such...Ch. 4.1 - Prob. 6PCh. 4.1 - If x=(2+i,3i) and y=(5,22i) in 2 find a vector z...Ch. 4.1 - If x=(5+i,0,12i,1+8i) and y=(3,i,i,3) in 4 find a...Ch. 4.1 - Verify the commutative law of addition of vectors...Ch. 4.1 - Prob. 10PCh. 4.1 - Prob. 11PCh. 4.1 - Prob. 12PCh. 4.1 - Show with example that if x in the first quadrant...Ch. 4.2 - For Questions a-j, decide if the given statement...Ch. 4.2 - True-False Review For items a-j, decide if the...Ch. 4.2 - For Questions a-j, decide if the given statement...Ch. 4.2 - For Questions a-j, decide if the given statement...Ch. 4.2 - For Questions a-j, decide if the given statement...Ch. 4.2 - For Questions a-j, decide if the given statement...Ch. 4.2 - For Questions a-j, decide if the given statement...Ch. 4.2 - PROBLEMS For Problems 1-14, determine whether the...Ch. 4.2 - PROBLEMS For Problems 1-14, determine whether the...Ch. 4.2 - PROBLEMS For Problems 1-14, determine whether the...Ch. 4.2 - PROBLEMS For Problems 1-14, determine whether the...Ch. 4.2 - Problems For Problems 1-14, determine whether the...Ch. 4.2 - PROBLEMS For Problems 1-14, determine whether the...Ch. 4.2 - PROBLEMS For Problems 1-14, determine whether the...Ch. 4.2 - PROBLEMS For Problems 1-14, determine whether the...Ch. 4.2 - PROBLEMS For Problems 1-14, determine whether the...Ch. 4.2 - PROBLEMS For Problems 1-14, determine whether the...Ch. 4.2 - PROBLEMS For Problems 1-14, determine whether the...Ch. 4.2 - PROBLEMS For Problems 1-14, determine whether the...Ch. 4.2 - PROBLEMS For Problems 1-14, determine whether the...Ch. 4.2 - PROBLEMS For Problems 1-14, determine whether the...Ch. 4.2 - We have defined the set 2={(x,y):x,y}, together...Ch. 4.2 - Determine the zero vector in the vector space...Ch. 4.2 - Generalize the previous exercise to find the zero...Ch. 4.2 - Determine the zero vector in the vector space...Ch. 4.2 - Generalize the previous exercise to find the zero...Ch. 4.2 - On +, the set of positive real numbers, define the...Ch. 4.2 - On 2, define the operations of addition and scalar...Ch. 4.2 - On 2, define the operations of addition and scalar...Ch. 4.2 - Prob. 23PCh. 4.2 - On M2(), define the operation of addition by...Ch. 4.2 - For Problems 2627, verify that the given set of...Ch. 4.2 - For Problems 2627, verify that the given set of...Ch. 4.2 - Is 3 a real vector space? Explain.Ch. 4.2 - Prob. 29PCh. 4.2 - Prob. 30PCh. 4.2 - Prob. 31PCh. 4.2 - Prove that Pn() is a vector space.Ch. 4.3 - For the problems a-h, decide if the given...Ch. 4.3 - Problems Let S={x3:x=(r2s,3r+s,s),r,s} aShow that...Ch. 4.3 - Problems Let S={x2:x=(2k,3k),k} aShow that S is a...Ch. 4.3 - Problems For Problems 322, express S in set...Ch. 4.3 - Problems For Problems 322, express S in set...Ch. 4.3 - Problems For Problems 322, express S in set...Ch. 4.3 - Problems For Problems 322, express S in set...Ch. 4.3 - Problems For Problems 322, express S in set...Ch. 4.3 - Problems For Problems 322, express S in set...Ch. 4.3 - Problems For Problems 322, express S in set...Ch. 4.3 - Problems For Problems 322, express S in set...Ch. 4.3 - Problems For Problems 322, express S in set...Ch. 4.3 - Problems For Problems 322, express S in set...Ch. 4.3 - Problems For Problems 322, express S in set...Ch. 4.3 - Problems For Problems 322, express S in set...Ch. 4.3 - Problems For Problems 322, express S in set...Ch. 4.3 - Problems For Problems 322, express S in set...Ch. 4.3 - Problems For Problems 322, express S in set...Ch. 4.3 - Problems For Problems 322, express S in set...Ch. 4.3 - Problems For Problems 322, express S in set...Ch. 4.3 - Problems For Problems 322, express S in set...Ch. 4.3 - Problems For Problems 322, express S in set...Ch. 4.3 - Problems For Problems 322, express S in set...Ch. 4.3 - Problems For Problems 2329, determine the null...Ch. 4.3 - Problems For Problems 2329, determine the null...Ch. 4.3 - Problems For Problems 2329, determine the null...Ch. 4.3 - Problems For Problems 2329, determine the null...Ch. 4.3 - Problems For Problems 2329, determine the null...Ch. 4.3 - Problems For Problems 2329, determine the null...Ch. 4.3 - Problems For Problems 2329, determine the null...Ch. 4.3 - Problems Show that the set of all solutions to the...Ch. 4.3 - Problems Let S1 and S2 be subspaces of vector...Ch. 4.4 - Prob. 1TFRCh. 4.4 - For question (a)(l), decide if the given statement...Ch. 4.4 - Problems For Problems 1-4, determine whether the...Ch. 4.4 - Problems For Problems 1-4, determine whether the...Ch. 4.4 - Problems For Problems 1-4, determine whether the...Ch. 4.4 - Problems For Problems 1-4, determine whether the...Ch. 4.4 - Problems Recall that three vectors . in 3 are...Ch. 4.4 - Problems Recall that three vectors . in 3 are...Ch. 4.4 - Problems Recall that three vectors . in 3 are...Ch. 4.4 - Problems Recall that three vectors . in 3 are...Ch. 4.4 - Problems Show that the set of vectors...Ch. 4.4 - Problems Show that the set of vectors...Ch. 4.4 - Problems Show that v1=(2,1), v2=(3,2) span 2 and...Ch. 4.4 - Problems Show that v1=(1,5), v2=(6,3) span 2, and...Ch. 4.4 - Problems Show that v1=(1,3,2), v2=(1,0,1),...Ch. 4.4 - Problems Show that v1=(1,3,2), v2=(1,2,1),...Ch. 4.4 - Problems Show that v1=(1,1), v2=(1,2), v3=(1,4)...Ch. 4.4 - Problems Let S be the subspace of 3 consisting of...Ch. 4.4 - Problems Let S be the subspace of 4 consisting of...Ch. 4.4 - Problems Let S be the subspace of M2() consisting...Ch. 4.4 - Problems Let S be the subset of M2() consisting of...Ch. 4.4 - Problems Let S be the subspace of M2() consisting...Ch. 4.4 - Let S be the subspace of M3() consisting of all 33...Ch. 4.4 - Problems Let S be the subspace of M3() consisting...Ch. 4.4 - Problems Let S be the subspace of 3 consisting of...Ch. 4.4 - Problems Let S be the subspace of P3() consisting...Ch. 4.4 - Problems For Problems 2533, determine a spanning...Ch. 4.4 - Problems For Problems 2533, determine a spanning...Ch. 4.4 - Problems For Problems 2533, determine a spanning...Ch. 4.4 - Problems For Problems 2533, determine a spanning...Ch. 4.4 - Problems For Problems 2533, determine a spanning...Ch. 4.4 - Problems For Problems 2533, determine a spanning...Ch. 4.4 - Problems For Problems 2533, determine a spanning...Ch. 4.4 - Problems For Problems 2533, determine a spanning...Ch. 4.4 - Problems For Problems 2533, determine a spanning...Ch. 4.4 - For Problems 3435, determine span {v1,v2} for the...Ch. 4.4 - For Problems 3435, determine span {v1,v2} for the...Ch. 4.4 - Problems Let S be the subspace of 3 spanned by the...Ch. 4.4 - Problems For Problems 3739, determine whether the...Ch. 4.4 - Prob. 38PCh. 4.4 - Problems For Problems 3739, determine whether the...Ch. 4.4 - Problems If p1(x)=x4 and p2(x)=x2x+3, determine...Ch. 4.4 - Problems Consider the vectors A1=[1120],...Ch. 4.4 - Problems Consider the vectors A1=[1213], A2=[2111]...Ch. 4.4 - Prob. 43PCh. 4.4 - Prob. 44PCh. 4.4 - Prob. 45PCh. 4.4 - Prob. 46PCh. 4.4 - Prob. 47PCh. 4.4 - Prob. 48PCh. 4.4 - Prove that span{v1,v2,v3}=span{v1,v2} if and only...Ch. 4.5 - For problem 1-10, determine whether the given set...Ch. 4.5 - For problem 1-10, determine whether the given set...Ch. 4.5 - For problem 1-10, determine whether the given set...Ch. 4.5 - For problem 1-10, determine whether the given set...Ch. 4.5 - For problem 1-10, determine whether the given set...Ch. 4.5 - For problem 1-10, determine whether the given set...Ch. 4.5 - For problem 1-10, determine whether the given set...Ch. 4.5 - For problem 1-10, determine whether the given set...Ch. 4.5 - For problem 1-10, determine whether the given set...Ch. 4.5 - For problem 1-10, determine whether the given set...Ch. 4.5 - Let v1=(1,2,3), v2=(4,5,6,), v3=(7,8,9), determine...Ch. 4.5 - Determine all the values of constant k for which...Ch. 4.5 - For Problem 14-15, determine all the values of...Ch. 4.5 - For Problems 14-15, determine all the values of...Ch. 4.5 - For Problem 16-18, determine whether the given set...Ch. 4.5 - For Problem 16-18, determine whether the given set...Ch. 4.5 - For Problem 16-18, determine whether the given set...Ch. 4.5 - For problem 19-22, determine whether the given set...Ch. 4.5 - For problem 19-22, determine whether the given set...Ch. 4.5 - For problem 19-22, determine whether the given set...Ch. 4.5 - For problem 19-22, determine whether the given set...Ch. 4.5 - Show that the vectors p1(x)=a+bx and p2(x)=c+dx...Ch. 4.5 - If f1(x)=cos2x, f2(x)=sin2x, f3(x)=cos2x,...Ch. 4.5 - For Problems 25-31, determine a linearly...Ch. 4.5 - Prob. 26PCh. 4.5 - For Problems, 25-31, determine a linearly...Ch. 4.5 - For Problems, 25-31, determine a linearly...Ch. 4.5 - For Problems, 25-31, determine a linearly...Ch. 4.5 - For Problems, 25-31, determine a linearly...Ch. 4.5 - For Problems 32-36, use the Wronskian to show that...Ch. 4.5 - Prob. 35PCh. 4.5 - Prob. 36PCh. 4.5 - For Problems 37-39, show that the Wronskian of the...Ch. 4.5 - Prob. 38PCh. 4.5 - Prob. 39PCh. 4.5 - Prob. 41PCh. 4.5 - a. Show that {1,x,x2,x3} is linearly independent...Ch. 4.5 - Prob. 45PCh. 4.5 - Problems If v1andv2 are vectors in a vector space...Ch. 4.5 - Prove from Definition 4.5.4 that if {v1,v2,...,vn}...Ch. 4.5 - Prob. 49PCh. 4.5 - Problems Generalizing the previous exercise prove...Ch. 4.5 - Problems Prove Theorem 4.5.2. Theorem 4.5.2.Let...Ch. 4.5 - Problems Prove Proposition 4.5.8. Let V be a...Ch. 4.5 - Problems Prove that if {v1,v2,......,vk} spans a...Ch. 4.5 - Prob. 54PCh. 4.6 - For Questions a-k, decide if the given statement...Ch. 4.6 - For Problems 1-7, determine whether the given set...Ch. 4.6 - For Problems 1-7, determine whether the given set...Ch. 4.6 - For Problems 1-7, determine whether the given set...Ch. 4.6 - For Problems 1-7, determine whether the given set...Ch. 4.6 - For Problems 1-7, determine whether the given set...Ch. 4.6 - For Problems 1-7, determine whether the given set...Ch. 4.6 - For Problems 1-7, determine whether the given set...Ch. 4.6 - Determine all values of the constant k for which...Ch. 4.6 - For Problems 9-14, determine whether the given set...Ch. 4.6 - For Problems 9-14, determine whether the given set...Ch. 4.6 - For Problems 9-14, determine whether the given set...Ch. 4.6 - For Problems 9-14, determine whether the given set...Ch. 4.6 - For Problems 9-14, determine whether the given set...Ch. 4.6 - For Problems 9-14, determine whether the given set...Ch. 4.6 - For Problems 15-18, determine whether the given...Ch. 4.6 - For Problems 15-18, determine whether the given...Ch. 4.6 - For Problems 15-18, determine whether the given...Ch. 4.6 - For Problems 15-18, determine whether the given...Ch. 4.6 - For Problems 19-23, find the dimension of the null...Ch. 4.6 - For Problems 19-23, find the dimension of the null...Ch. 4.6 - For Problems 19-23, find the dimension of the null...Ch. 4.6 - For Problems 19-23, find the dimension of the null...Ch. 4.6 - For Problems 19-23, find the dimension of the null...Ch. 4.6 - Let S be the subspace of 3 that consists of all...Ch. 4.6 - Let S be the subspace of 3 consisting of all...Ch. 4.6 - Determine a basis S for P3(), and hence, prove...Ch. 4.6 - Determine a basis S for P3() whose elements all...Ch. 4.6 - Let S be the subspace of M2() consisting of all 22...Ch. 4.6 - Let S be the subspace of M2() consisting of all 22...Ch. 4.6 - Let S be the subspace of 3 spanned by the vectors...Ch. 4.6 - Let S be the vectors space consisting of the set...Ch. 4.6 - Determine a basis for the subspace of M2() spanned...Ch. 4.6 - Let v1=(1,1) and v2=(1,1). a Show that {v1,v2}...Ch. 4.6 - Prob. 34PCh. 4.6 - Let v1=(0,6,3), v2=(3,0,3), and v3=(6,3,0). Show...Ch. 4.6 - Prob. 36PCh. 4.6 - Let p1(x)=1+x, p2(x)=x+x2, and p3(x)=1+2x2. Show...Ch. 4.6 - Prob. 38PCh. 4.6 - Prob. 39PCh. 4.6 - Prob. 40PCh. 4.6 - Prob. 43PCh. 4.6 - Prob. 44PCh. 4.6 - For Problems 45-47, a subspace S of a vector space...Ch. 4.6 - For Problems 45-47, a subspace S of a vector space...Ch. 4.6 - For problems 45-47, a subspace S of a vector space...Ch. 4.6 - For Problems 48-50, determine a basis for the...Ch. 4.6 - For Problems 48-50, determine a basis for the...Ch. 4.6 - For Problems 48-50, determine a basis for the...Ch. 4.6 - Prob. 51PCh. 4.6 - Prob. 52PCh. 4.6 - Prob. 53PCh. 4.6 - Prob. 54PCh. 4.7 - True-False Review For Questions a h, decide if...Ch. 4.7 - Problems For Problems 1-14, determine the...Ch. 4.7 - PROBLEMS For Problems 1-14, determine the...Ch. 4.7 - PROBLEMS For Problems 1-14, determine the...Ch. 4.7 - Prob. 4PCh. 4.7 - PROBLEMS For Problems 1-14, determine the...Ch. 4.7 - PROBLEMS For Problems 1-14, determine the...Ch. 4.7 - PROBLEMS For Problems 1-14, determine the...Ch. 4.7 - PROBLEMS For Problems 1-14, determine the...Ch. 4.7 - PROBLEMS For Problems 1-14, determine the...Ch. 4.7 - PROBLEMS For Problems 1-14, determine the...Ch. 4.7 - PROBLEMS For Problems 1-14, determine the...Ch. 4.7 - PROBLEMS For Problems 1-14, determine the...Ch. 4.7 - PROBLEMS For Problems 1-14, determine the...Ch. 4.7 - Prob. 14PCh. 4.7 - PROBLEMS Let v1=(0,6,3), v2=(3,0,3), and...Ch. 4.7 - PROBLEMS Let p1(x)=1+x, p2(x)=x+x2, and...Ch. 4.7 - PROBLEMS For Problems 17-26, find the...Ch. 4.7 - PROBLEMS For Problems 17-26, find the...Ch. 4.7 - PROBLEMS For Problems 17-26, find the...Ch. 4.7 - PROBLEMS For Problems 17-26, find the...Ch. 4.7 - PROBLEMS For Problems 17-26, find the...Ch. 4.7 - PROBLEMS For Problems 17-26, find the...Ch. 4.7 - PROBLEMS For Problems 17-26, find the...Ch. 4.7 - PROBLEMS For Problems 17-26, find the...Ch. 4.7 - PROBLEMS For Problems 17-26, find the...Ch. 4.7 - PROBLEMS For Problems 17-26, find the...Ch. 4.7 - PROBLEMS For Problems 27-32, find the...Ch. 4.7 - PROBLEMS For Problems 27-32, find the...Ch. 4.7 - Prob. 29PCh. 4.7 - PROBLEMS For Problems 27-32, find the...Ch. 4.7 - PROBLEMS For Problems 27-32, find the...Ch. 4.7 - Prob. 32PCh. 4.7 - PROBLEMS For Problems 33-37, verify Equation...Ch. 4.7 - Prob. 34PCh. 4.7 - Prob. 35PCh. 4.7 - Prob. 36PCh. 4.7 - Prob. 37PCh. 4.7 - Prob. 38PCh. 4.7 - PROBLEMS Prove that if every vector v in a vector...Ch. 4.7 - Show that if B is a basis for a finite-dimensional...Ch. 4.8 - Prob. 1TFRCh. 4.8 - For Problem 12, a determine a basis for...Ch. 4.8 - For Problem 12, a determine a basis for...Ch. 4.8 - For Problem 39, a find n such that rowspace(A) is...Ch. 4.8 - For Problem 39, a find n such that rowspace(A) is...Ch. 4.8 - For Problem 39, a find n such that rowspace(A) is...Ch. 4.8 - For Problem 39, a find n such that rowspace(A) is...Ch. 4.8 - For Problem 39, a find n such that rowspace(A) is...Ch. 4.8 - For Problem 39, a find n such that rowspace(A) is...Ch. 4.8 - For Problem 39, a find n such that rowspace(A) is...Ch. 4.8 - For Problem 1013, determine a basis for the...Ch. 4.8 - For Problem 1013, determine a basis for the...Ch. 4.8 - For Problem 1013, determine a basis for the...Ch. 4.8 - For Problem 1013, determine a basis for the...Ch. 4.8 - Let A=[124511213713]. a Find a basis for...Ch. 4.8 - Prob. 15PCh. 4.8 - Give examples to show how each type of elementary...Ch. 4.8 - Let A be an mn matrix with...Ch. 4.8 - Let A be an nn matrix with...Ch. 4.9 - Prob. 1TFRCh. 4.9 - For problem 1-5, determine the null space of A and...Ch. 4.9 - Prob. 2PCh. 4.9 - For problem 1-5, determine the null space of A and...Ch. 4.9 - For problem 1-5, determine the null space of A and...Ch. 4.9 - For problem 1-5, determine the null space of A and...Ch. 4.9 - For problem 6-9, determine the nullity of A by...Ch. 4.9 - For problem 6-9, determine the nullity of A by...Ch. 4.9 - For problem 6-9, determine the nullity of A by...Ch. 4.9 - For problem 6-9, determine the nullity of A by...Ch. 4.9 - For problem 10-13, determine the solution set to...Ch. 4.9 - For problem 10-13, determine the solution set to...Ch. 4.9 - For problem 10-13, determine the solution set to...Ch. 4.9 - Prob. 13PCh. 4.9 - Problems Show that a 37 matrix A with nullity(A)=4...Ch. 4.9 - Problems Prove that a 64 matrix A with...Ch. 4.9 - Problems Prove that if rowspace(A)=nullspace(A),...Ch. 4.9 - Prob. 17PCh. 4.9 - Problems Show that 38 matrix A must have...Ch. 4.9 - Prob. 19PCh. 4.11 - Additional Problems For problem 1-2, let r and s...Ch. 4.11 - Additional Problems For problem 1-2, let r and s...Ch. 4.11 - Additional Problems For problem 3-12, determine...Ch. 4.11 - For Problems 3-12, determine whether the given set...Ch. 4.11 - Prob. 8APCh. 4.11 - Prob. 9APCh. 4.11 - Prob. 10APCh. 4.11 - Prob. 23APCh. 4.11 - Prob. 24APCh. 4.11 - Prob. 25APCh. 4.11 - Prob. 26APCh. 4.11 - Prob. 27APCh. 4.11 - Prob. 28APCh. 4.11 - Prob. 29APCh. 4.11 - Prob. 30APCh. 4.11 - Prob. 31APCh. 4.11 - Additional Problems Prove that if {v1,v2,v3} is...Ch. 4.11 - Let A be an mn matrix. Show that the columns of A...Ch. 4.11 - Let (V,+v,v) and (W,+w,w) be vector spaces and...Ch. 4.11 - Additional Problem Show that a basis for P3() need...Ch. 4.11 - Prob. 38APCh. 4.11 - Prob. 39APCh. 4.11 - Prob. 40APCh. 4.11 - Prob. 41APCh. 4.11 - Prob. 42APCh. 4.11 - Additional Problems For Problem 40-43, find a...Ch. 4.11 - Prob. 44AP

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