Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)
5th Edition
ISBN: 9780134689531
Author: Lee Johnson, Dean Riess, Jimmy Arnold
Publisher: PEARSON
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Chapter 5.3, Problem 2E
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Chapter 5 Solutions
Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)
Ch. 5.2 - For u,v and w given in given Exercises 13,...Ch. 5.2 - For u,v and w given in given Exercises 13,...Ch. 5.2 - For u,v and w given in given Exercises 13,...Ch. 5.2 - Prob. 4ECh. 5.2 - Prob. 5ECh. 5.2 - In Exercises 6-11, the given set is a subset of a...Ch. 5.2 - Prob. 7ECh. 5.2 - In Exercises 6-11, the given set is a subset of a...Ch. 5.2 - In Exercises 6-11, the given set is a subset of a...Ch. 5.2 - In Exercises 6-11, the given set is a subset of a...
Ch. 5.2 - Prob. 11ECh. 5.2 - Prob. 12ECh. 5.2 - Prob. 13ECh. 5.2 - Prob. 14ECh. 5.2 - Prob. 15ECh. 5.2 - Prob. 16ECh. 5.2 - Let Q denote the set of all (22) nonsingular...Ch. 5.2 - Let Q denote the set of all (22) singular matrices...Ch. 5.2 - Let Q denote the set of all (22) symmetric...Ch. 5.2 - Prove the cancellation laws for vector addition.Ch. 5.2 - Prove property 2 of Theorem 1. Hint: See the proof...Ch. 5.2 - Prove property 3 of Theorem 1. Hint: Note that...Ch. 5.2 - Prove property 5 of Theorem 1. If a0 then multiply...Ch. 5.2 - Prob. 24ECh. 5.2 - In Exercise s 25-29, the given set is a subset of...Ch. 5.2 - In Exercises 2529, the given set is a subset of...Ch. 5.2 - Prob. 27ECh. 5.2 - Prob. 28ECh. 5.2 - In Exercises 2529, the given set is a subset of...Ch. 5.2 - Prob. 30ECh. 5.2 - The following are subsets of vector space C2[1,1]....Ch. 5.2 - Prob. 32ECh. 5.2 - Let F(R) denote the set of all real valued...Ch. 5.2 - Let V={x:x=[x1x2],wherex1andx2areinR}. For u and v...Ch. 5.2 - Let, V={x:x=[x1x2],wherex1andx2areinR}. For u and...Ch. 5.2 - Let V={x:x=[x1x2],wherex20}. For u and v in V and...Ch. 5.3 - Let V be the vector space of all (23) matrices....Ch. 5.3 - Let V be the vector space of all (23) matrices....Ch. 5.3 - Let V be the vector space of all (23) matrices....Ch. 5.3 - Let V be the vector space of all (23) matrices....Ch. 5.3 - In Exercises 58, which of the given subsets of P2...Ch. 5.3 - Prob. 6ECh. 5.3 - In Exercises 58, which of the given subsets of P2...Ch. 5.3 - In Exercises 58, which of the given subsets of P2...Ch. 5.3 - In Exercises 912, which of the given subsets of...Ch. 5.3 - In Exercises 912, which of the given subsets of...Ch. 5.3 - In Exercises 912, which of the given subsets of...Ch. 5.3 - Prob. 12ECh. 5.3 - In Exercises 1316, which of the given subsets of...Ch. 5.3 - In Exercises 1316, which of the given subsets of...Ch. 5.3 - Prob. 15ECh. 5.3 - Prob. 16ECh. 5.3 - In Exercises 1721, express the given vector as a...Ch. 5.3 - In Exercises 1721, express the given vector as a...Ch. 5.3 - In Exercises 1721, express the given vector as a...Ch. 5.3 - In Exercises 1721, express the given vector as a...Ch. 5.3 - In Exercises 1721, express the given vector as a...Ch. 5.3 - Let V be the vector space of all (22) matrices....Ch. 5.3 - Let W be the subset of P3 defined by...Ch. 5.3 - Let W be the subset of P3 defined by...Ch. 5.3 - Find a spanning set for each of the subsets that...Ch. 5.3 - Show that the set W of all symmetric (33) matrices...Ch. 5.3 - The trace of an (nn) matrix A=(aij), denoted...Ch. 5.3 - Prob. 28ECh. 5.3 - Prob. 29ECh. 5.3 - Prob. 30ECh. 5.3 - Let V be the set of all (33) upper-triangular...Ch. 5.3 - Prob. 32ECh. 5.3 - Let A be an arbitrary matrix in the vector space...Ch. 5.4 - In exercise 14, W is a subspace of the vector...Ch. 5.4 - In exercise 14, W is a subspace of the vector...Ch. 5.4 - In exercise 14, W is a subspace of the vector...Ch. 5.4 - In exercise 1-4, W is a subspace of the vector...Ch. 5.4 - In exercise 58, W is a subspace of P2. In each...Ch. 5.4 - In Exercises 58, W is a subspace of P2. In each...Ch. 5.4 - In Exercises 58, W is a subspace of P2. In each...Ch. 5.4 - In Exercises 58, W is a subspace of P2. In each...Ch. 5.4 - Find a basis for the subspace V of P4, where...Ch. 5.4 - Prove that the set of all real (22) symmetric...Ch. 5.4 - Let V be the vector space of all (22) real...Ch. 5.4 - With respect to the basis B={1,x,x2} for P2, find...Ch. 5.4 - With respect to the basis B={E11,E12,E21,E22} for...Ch. 5.4 - Prove that {1,x,x2,......xn} is a linearly...Ch. 5.4 - In Exercise 1517, use the basis B of Exercise 11...Ch. 5.4 - In Exercise 1517, use the basis B of Exercise 11...Ch. 5.4 - In Exercise 1517, use the basis B of Exercise 11...Ch. 5.4 - In Exercise 1821, use Exercise 14 and property 2...Ch. 5.4 - In Exercise 1821, use Exercise 14 and property 2...Ch. 5.4 - In Exercise 1821, use Exercise 14 and property 2...Ch. 5.4 - In Exercise 1821, use Exercise 14 and property 2...Ch. 5.4 - 22. In P2, let S={p1(x),p2(x),p3(x),p4(x)}, where...Ch. 5.4 - Let S be the subset of P2 given in Exercise 22....Ch. 5.4 - Let V be the vector space of all (22) matrices and...Ch. 5.4 - Let V and S be as in Exercise 24. Find a subset of...Ch. 5.4 - In P2, let Q={p1(x),p2(x),p3(x)}, Where...Ch. 5.4 - Let Q be the basis for P2 given in Exercise 26....Ch. 5.4 - Let Q be the basis for P2 given in Exercise 26....Ch. 5.4 - In the vector space V of (22) matrices, let...Ch. 5.4 - With V and Q as in Exercise 29, find [A]Q for,...Ch. 5.4 - With V and Q as in Exercise 29, find [A]Q for,...Ch. 5.4 - Give an alternate proof that {1,x,x2} is a...Ch. 5.4 - The set {sinx,cosx} is a subset of the vector...Ch. 5.4 - In Exercises 34 and 35, V is the set of...Ch. 5.4 - In Exercises 34 and 35, V is the set of...Ch. 5.4 - Prob. 36ECh. 5.4 - Prob. 37ECh. 5.4 - Use Exercise 37 to obtain necessary and sufficient...Ch. 5.5 - 1.Let V be the set of all real (33) matrices, and...Ch. 5.5 - Prob. 2ECh. 5.5 - Prob. 3ECh. 5.5 - Prob. 4ECh. 5.5 - Recall that a square matrix A is called the skew...Ch. 5.5 - Prob. 6ECh. 5.5 - Prob. 7ECh. 5.5 - In Exercises 813, a subset S of vector space V is...Ch. 5.5 - In Exercises 813, a subset S of vector space V is...Ch. 5.5 - In Exercises 813, a subset S of vector space V is...Ch. 5.5 - In Exercises 813, a subset S of vector space V is...Ch. 5.5 - In Exercises 813, a subset S of vector space V is...Ch. 5.5 - In Exercises 813, a subset S of vector space V is...Ch. 5.5 - 14. Let W be the subspace of C[,] consisting of...Ch. 5.5 - Let V denote the set of all infinite sequences of...Ch. 5.5 - Prob. 16ECh. 5.5 - Let W be a subspace of a finite-dimensional vector...Ch. 5.5 - Prob. 18ECh. 5.5 - Prob. 19ECh. 5.5 - Prob. 20ECh. 5.5 - Prob. 21ECh. 5.5 - Prob. 22ECh. 5.5 - By Theorem 5 of Section 5.4, an (nn) transition...Ch. 5.5 - Prob. 24ECh. 5.6 - Prove that x,y=4x1y1+x2y2 is an inner product on...Ch. 5.6 - Prob. 2ECh. 5.6 - A real (nn) symmetric matrix A is called positive...Ch. 5.6 - Prove that the following symmetric matrix A is...Ch. 5.6 - Prob. 5ECh. 5.6 - Prob. 6ECh. 5.6 - Prob. 7ECh. 5.6 - Prob. 8ECh. 5.6 - Prob. 9ECh. 5.6 - In P2, let p(x)=1+2x+x2 and q(x)=1x+2x2. Using the...Ch. 5.6 - Prob. 11ECh. 5.6 - Prob. 12ECh. 5.6 - Prob. 13ECh. 5.6 - Prob. 14ECh. 5.6 - Let {u1,u2} be the orthogonal basis for R2...Ch. 5.6 - Prob. 16ECh. 5.6 - Prob. 17ECh. 5.6 - Prob. 18ECh. 5.6 - Prob. 19ECh. 5.6 - Prob. 20ECh. 5.6 - Prob. 21ECh. 5.6 - Prob. 22ECh. 5.6 - Prob. 23ECh. 5.6 - Prob. 24ECh. 5.6 - Prob. 25ECh. 5.6 - Prob. 26ECh. 5.6 - Prob. 27ECh. 5.6 - Prob. 28ECh. 5.6 - A sequence of orthogonal polynomials usually...Ch. 5.6 - Prob. 30ECh. 5.6 - Show that if A is a real (nn) matrix and if the...Ch. 5.7 - In Exercises 14, V is the vector space of all (22)...Ch. 5.7 - In Exercises 14, V is the vector space of all (22)...Ch. 5.7 - In Exercises 14, V is the vector space of all (22)...Ch. 5.7 - In Exercises 14, V is the vector space of all (22)...Ch. 5.7 - In Exercises 58, determine whether T is a linear...Ch. 5.7 - In Exercises 58, determine whether T is a linear...Ch. 5.7 - In Exercises 58, determine whether T is a linear...Ch. 5.7 - In Exercises 58, determine whether T is a linear...Ch. 5.7 - Suppose that T:P2P3 is a linear transformation,...Ch. 5.7 - 10. Suppose that T:P2P4 is a linear...Ch. 5.7 - Let V be the set of all (22) matrices and suppose...Ch. 5.7 - With V as in Exercise 11, define T:VR2 by...Ch. 5.7 - Let T:P4P2 be the linear transformation defined by...Ch. 5.7 - Define T:P4P3 by...Ch. 5.7 - Identify N(T) and R(T) for the linear...Ch. 5.7 - Identify N(T) and R(T) for the linear...Ch. 5.7 - Prob. 17ECh. 5.7 - Prob. 18ECh. 5.7 - Suppose that T:P4P2 is a linear transformation....Ch. 5.7 - Prob. 21ECh. 5.7 - Prob. 22ECh. 5.7 - Prob. 23ECh. 5.7 - Prob. 24ECh. 5.7 - Prob. 25ECh. 5.7 - Prob. 26ECh. 5.7 - Let V be the vector space of all (22) matrices and...Ch. 5.8 - In Exercises 16, the linear transformations S,T,...Ch. 5.8 - In Exercises 16, the linear transformations S,T,...Ch. 5.8 - In Exercises 16, the linear transformations S,T,...Ch. 5.8 - In Exercises 16, the linear transformations S,T,...Ch. 5.8 - In Exercises 16, the linear transformations S,T,...Ch. 5.8 - In Exercises 16, the linear transformations S,T,...Ch. 5.8 - 7. The functions ex,e2x and e3x are linearly...Ch. 5.8 - Let V be the subspace of C[0,1] defined by...Ch. 5.8 - Let V be the vector space of all 22 matrices and...Ch. 5.8 - Let V be the vector space of all (22) matrices,...Ch. 5.8 - Prob. 11ECh. 5.8 - Let U be the vector space of all (22) symmetric...Ch. 5.8 - Prob. 13ECh. 5.8 - Prob. 14ECh. 5.8 - Prob. 15ECh. 5.8 - Prob. 16ECh. 5.8 - Prob. 17ECh. 5.8 - Let S:UV and T:VW be linear transformations. a...Ch. 5.8 - Prob. 19ECh. 5.8 - Prob. 20ECh. 5.8 - Prob. 21ECh. 5.8 - Prob. 22ECh. 5.8 - Prob. 23ECh. 5.8 - Prob. 24ECh. 5.8 - Prob. 25ECh. 5.8 - Prob. 26ECh. 5.8 - Prob. 27ECh. 5.8 - Prob. 28ECh. 5.8 - Prob. 29ECh. 5.9 - In Exercises 110, the linear transformations S,T,H...Ch. 5.9 - In Exercises 110, the linear transformations S,T,H...Ch. 5.9 - In Exercises 110, the linear transformations S,T,H...Ch. 5.9 - In Exercises 110, the linear transformations S,T,H...Ch. 5.9 - In Exercises 110, the linear transformations S,T,H...Ch. 5.9 - In Exercises 110, the linear transformations S,T,H...Ch. 5.9 - In Exercises 110, the linear transformations S,T,H...Ch. 5.9 - In Exercises 110, the linear transformations S,T,H...Ch. 5.9 - In Exercises 110, the linear transformations S,T,H...Ch. 5.9 - In Exercises 110, the linear transformations S,T,H...Ch. 5.9 - Let T:VV be the linear transformation defined in...Ch. 5.9 - Let T:VV be the linear transformation defined in...Ch. 5.9 - Let V be the vector space of (22) matrices and...Ch. 5.9 - Let S:P2P3 be given by S(p)=x3px2p+3p. Find the...Ch. 5.9 - Let S be the transformation in Exercise 14, let...Ch. 5.9 - Let S be the transformation in Exercise 14, let...Ch. 5.9 - Let T:P2R3 be given by T(p)=[p(0)3p(1)p(1)+p(0)]....Ch. 5.9 - Find the representation for the transformation in...Ch. 5.9 - Let T:VV be a linear transformation, where...Ch. 5.9 - Let T:R3R2 be given by T(x)=Ax, where A=[121304]....Ch. 5.9 - Let T:P2P2 be defined by...Ch. 5.9 - Let T be the linear transformation defined in...Ch. 5.9 - Let T be the linear transformation defined in...Ch. 5.9 - Prob. 24ECh. 5.9 - Prob. 25ECh. 5.9 - Prob. 26ECh. 5.9 - Prob. 27ECh. 5.9 - Prob. 28ECh. 5.9 - Prob. 29ECh. 5.9 - Prob. 30ECh. 5.9 - In Exercise 31 and 32, Q is the (34) matrix given...Ch. 5.9 - Prob. 32ECh. 5.9 - Complete the proof of theorem 21 by showing that...Ch. 5.10 - Let T:R2R2 is defined by T([x1x2])=[2x1+x2x1+2x2]...Ch. 5.10 - Let T:P2P2 is defined by...Ch. 5.10 - Prob. 3ECh. 5.10 - Prob. 4ECh. 5.10 - Prob. 5ECh. 5.10 - Prob. 6ECh. 5.10 - Prob. 7ECh. 5.10 - Repeat Exercise 7 for the basis vectors w1=[43],...Ch. 5.10 - Prob. 9ECh. 5.10 - Represent the following quadratic polynomials in...Ch. 5.10 - Prob. 11ECh. 5.10 - Let T:P2P2 is a linear transformation defined in...Ch. 5.10 - Prob. 13ECh. 5.10 - In Exercises 14-16, proceed through the following...Ch. 5.10 - In Exercises 14-16, proceed through the following...Ch. 5.10 - In Exercises 14-16, proceed through the following...Ch. 5.10 - Prob. 17ECh. 5.10 - Prob. 18ECh. 5.10 - Prob. 19ECh. 5.10 - Prob. 20ECh. 5.10 - Prob. 21ECh. 5.SE - Let V be the set of all 2x2 matrices with Real...Ch. 5.SE - Prob. 2SECh. 5.SE - Prob. 3SECh. 5.SE - Prob. 4SECh. 5.SE - Prob. 5SECh. 5.SE - Prob. 6SECh. 5.SE - Prob. 7SECh. 5.SE - In Exercises 7-11, use the fact that the matrix...Ch. 5.SE - Prob. 9SECh. 5.SE - In Exercises 7-11, use the fact that the matrix...Ch. 5.SE - In Exercise 7-11, Use the fact that the matrix...Ch. 5.SE - Show that there is a linear transformations T:R2P2...Ch. 5.SE - Prob. 13SECh. 5.SE - Let V be the vector space for all (22) matrices,...Ch. 5.CE - In Exercise 1-10, answer true or false. Justify...Ch. 5.CE - In Exercise 1-10, answer true or false. Justify...Ch. 5.CE - Prob. 3CECh. 5.CE - In Exercise 1-10, answer true or false. Justify...Ch. 5.CE - Prob. 5CECh. 5.CE - In Exercise 1-10, answer true or false. Justify...Ch. 5.CE - Prob. 7CECh. 5.CE - In Exercise 1-10, answer true or false. Justify...Ch. 5.CE - In Exercise 1-10, answer true or false. Justify...Ch. 5.CE - In Exercise 1-10, answer true or false. Justify...Ch. 5.CE - In Exercise 11-19, give a brief answer. Let W be a...Ch. 5.CE - In Exercise 11-19, give a brief answer. Let W be a...Ch. 5.CE - Prob. 13CECh. 5.CE - In Exercise 11-19, give a brief answer. Give...Ch. 5.CE - In Exercise 11-19, give a brief answer. If U and W...Ch. 5.CE - In Exercise 11-19, give a brief answer. Let...Ch. 5.CE - In Exercise 11-19, give a brief answer. Let...Ch. 5.CE - Let T:VW be a linear transformation. a.If T is one...Ch. 5.CE - Prob. 19CE
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- Consider an mn matrix A and an np matrix B. Show that the row vectors of AB are in the row space of B and the column vectors of AB are in the column space of A.arrow_forwardLet u, v, and w be any three vectors from a vector space V. Determine whether the set of vectors {vu,wv,uw} is linearly independent or linearly dependent.arrow_forwardIn Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=3, W={[aba]}arrow_forward
- In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=3, W={[aba+b+1]}arrow_forwardIn Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=3, W={[a0a]}arrow_forwardIn Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=M22,W={[abcd]:adbc}arrow_forward
- In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=Mnn, W is the set of diagonal nn matricesarrow_forwardLet A and B be square matrices of order n satisfying, Ax=Bx for all x in all Rn. a Find the rank and nullity of AB. b Show that matrices A and B must be identical.arrow_forwardLet A be an mn matrix where mn whose rank is r. a What is the largest value r can be? b How many vectors are in a basis for the row space of A? c How many vectors are in a basis for the column space of A? d Which vector space Rk has the row space as a subspace? e Which vector space Rk has the column space as a subspace?arrow_forward
- Let v1, v2, and v3 be three linearly independent vectors in a vector space V. Is the set {v12v2,2v23v3,3v3v1} linearly dependent or linearly independent? Explain.arrow_forwardIn Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=M22,W={[abb2a]}arrow_forwardTake this test to review the material in Chapters 4 and 5. After you are finished, check your work against the answers in the back of the book. Prove that the set of all singular 33 matrices is not a vector space.arrow_forward
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