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Math Algebra Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)

Prove that 〈 x , y 〉 = 4 x 1 y 1 + x 2 y 2 is an inner product on R 2 , where x = [ x 1 x 2 ] and y = [ y 1 y 2 ]

Prove that 〈 x , y 〉 = 4 x 1 y 1 + x 2 y 2 is an inner product on R 2 , where x = [ x 1 x 2 ] and y = [ y 1 y 2 ]

Solution Summary: The author explains that the pair of vectors langle x,yrangle is an inner product on the vector space R2.

Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)

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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Textbook Question

Chapter 5.6, Problem 1E

Prove that 〈 x , y 〉 = 4 x 1 y 1 + x 2 y 2 is an inner product on R 2 , where

x = [ x 1 x 2 ] and y = [ y 1 y 2 ]

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Chapter 5.5, Problem 24E

Chapter 5.6, Problem 2E

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Verify that the operation {x, y} = x1y1 − x1y2 − x2y1 + 3x2y2 where x = (x1, x2) and y = (y1, y2) is an inner product in R2.

. Let V be the set of all pairs (x,y) of real numbers together with the following operations: (x1,y1) (x2,y2) = (4x1 + x2 − 4, Y1 + 3 y2 − 3) c(x,y) = (cx- c+1, cy − c + 1). (a) Show that scalar multiplication distributes over vector addition, that is: c((x1,y1)(x2,y2)) = (c○ (x1,y1)) + (c○ (x2,y2)). (b) Explain why V nonetheless is not a vector space by showing that a vector space property does not hold for this set with these operations.

(3) Let V be R2, the set of all ordered pairs (x, y) of real numbers. Define an operation of "addition" by (u, v) (x, y) = (u+x+1,v+y+1) for all (u, v) and (x, y) in V. Define an operation of "scalar multiplication" by a(r, y) = (ar, ay) for all a ER and (x, y) = V. Under the operations and the set V is not a vector space. The vector space axioms (see 5.1.1 (1)-(10)) which fail to hold are and

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. 4E Ch. 5.2 - Prob. 5E Ch. 5.2 - In Exercises 6-11, the given set is a subset of a...Ch. 5.2 - Prob. 7E 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 - In Exercises 6-11, the given set is a subset of a...

Ch. 5.2 - Prob. 11E Ch. 5.2 - Prob. 12E Ch. 5.2 - Prob. 13E Ch. 5.2 - Prob. 14E Ch. 5.2 - Prob. 15E Ch. 5.2 - Prob. 16E Ch. 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. 24E Ch. 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. 27E Ch. 5.2 - Prob. 28E Ch. 5.2 - In Exercises 2529, the given set is a subset of...Ch. 5.2 - Prob. 30E Ch. 5.2 - The following are subsets of vector space C2[1,1]....Ch. 5.2 - Prob. 32E Ch. 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. 6E Ch. 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. 12E Ch. 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. 15E Ch. 5.3 - Prob. 16E 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 - 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. 28E Ch. 5.3 - Prob. 29E Ch. 5.3 - Prob. 30E Ch. 5.3 - Let V be the set of all (33) upper-triangular...Ch. 5.3 - Prob. 32E Ch. 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. 36E Ch. 5.4 - Prob. 37E Ch. 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. 2E Ch. 5.5 - Prob. 3E Ch. 5.5 - Prob. 4E Ch. 5.5 - Recall that a square matrix A is called the skew...Ch. 5.5 - Prob. 6E Ch. 5.5 - Prob. 7E 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 - 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. 16E Ch. 5.5 - Let W be a subspace of a finite-dimensional vector...Ch. 5.5 - Prob. 18E Ch. 5.5 - Prob. 19E Ch. 5.5 - Prob. 20E Ch. 5.5 - Prob. 21E Ch. 5.5 - Prob. 22E Ch. 5.5 - By Theorem 5 of Section 5.4, an (nn) transition...Ch. 5.5 - Prob. 24E Ch. 5.6 - Prove that x,y=4x1y1+x2y2 is an inner product on...Ch. 5.6 - Prob. 2E Ch. 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. 5E Ch. 5.6 - Prob. 6E Ch. 5.6 - Prob. 7E Ch. 5.6 - Prob. 8E Ch. 5.6 - Prob. 9E Ch. 5.6 - In P2, let p(x)=1+2x+x2 and q(x)=1x+2x2. Using the...Ch. 5.6 - Prob. 11E Ch. 5.6 - Prob. 12E Ch. 5.6 - Prob. 13E Ch. 5.6 - Prob. 14E Ch. 5.6 - Let {u1,u2} be the orthogonal basis for R2...Ch. 5.6 - Prob. 16E Ch. 5.6 - Prob. 17E Ch. 5.6 - Prob. 18E Ch. 5.6 - Prob. 19E Ch. 5.6 - Prob. 20E Ch. 5.6 - Prob. 21E Ch. 5.6 - Prob. 22E Ch. 5.6 - Prob. 23E Ch. 5.6 - Prob. 24E Ch. 5.6 - Prob. 25E Ch. 5.6 - Prob. 26E Ch. 5.6 - Prob. 27E Ch. 5.6 - Prob. 28E Ch. 5.6 - A sequence of orthogonal polynomials usually...Ch. 5.6 - Prob. 30E Ch. 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. 17E Ch. 5.7 - Prob. 18E Ch. 5.7 - Suppose that T:P4P2 is a linear transformation....Ch. 5.7 - Prob. 21E Ch. 5.7 - Prob. 22E Ch. 5.7 - Prob. 23E Ch. 5.7 - Prob. 24E Ch. 5.7 - Prob. 25E Ch. 5.7 - Prob. 26E Ch. 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. 11E Ch. 5.8 - Let U be the vector space of all (22) symmetric...Ch. 5.8 - Prob. 13E Ch. 5.8 - Prob. 14E Ch. 5.8 - Prob. 15E Ch. 5.8 - Prob. 16E Ch. 5.8 - Prob. 17E Ch. 5.8 - Let S:UV and T:VW be linear transformations. a...Ch. 5.8 - Prob. 19E Ch. 5.8 - Prob. 20E Ch. 5.8 - Prob. 21E Ch. 5.8 - Prob. 22E Ch. 5.8 - Prob. 23E Ch. 5.8 - Prob. 24E Ch. 5.8 - Prob. 25E Ch. 5.8 - Prob. 26E Ch. 5.8 - Prob. 27E Ch. 5.8 - Prob. 28E Ch. 5.8 - Prob. 29E 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 - 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. 24E Ch. 5.9 - Prob. 25E Ch. 5.9 - Prob. 26E Ch. 5.9 - Prob. 27E Ch. 5.9 - Prob. 28E Ch. 5.9 - Prob. 29E Ch. 5.9 - Prob. 30E Ch. 5.9 - In Exercise 31 and 32, Q is the (34) matrix given...Ch. 5.9 - Prob. 32E Ch. 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. 3E Ch. 5.10 - Prob. 4E Ch. 5.10 - Prob. 5E Ch. 5.10 - Prob. 6E Ch. 5.10 - Prob. 7E Ch. 5.10 - Repeat Exercise 7 for the basis vectors w1=[43],...Ch. 5.10 - Prob. 9E Ch. 5.10 - Represent the following quadratic polynomials in...Ch. 5.10 - Prob. 11E Ch. 5.10 - Let T:P2P2 is a linear transformation defined in...Ch. 5.10 - Prob. 13E 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 - In Exercises 14-16, proceed through the following...Ch. 5.10 - Prob. 17E Ch. 5.10 - Prob. 18E Ch. 5.10 - Prob. 19E Ch. 5.10 - Prob. 20E Ch. 5.10 - Prob. 21E Ch. 5.SE - Let V be the set of all 2x2 matrices with Real...Ch. 5.SE - Prob. 2SE Ch. 5.SE - Prob. 3SE Ch. 5.SE - Prob. 4SE Ch. 5.SE - Prob. 5SE Ch. 5.SE - Prob. 6SE Ch. 5.SE - Prob. 7SE Ch. 5.SE - In Exercises 7-11, use the fact that the matrix...Ch. 5.SE - Prob. 9SE Ch. 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. 13SE Ch. 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. 3CE Ch. 5.CE - In Exercise 1-10, answer true or false. Justify...Ch. 5.CE - Prob. 5CE Ch. 5.CE - In Exercise 1-10, answer true or false. Justify...Ch. 5.CE - Prob. 7CE 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 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. 13CE Ch. 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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