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Math Algebra Elementary Linear Algebra (MindTap Course List)

Orthogonal and Orthonormal Sets In Exercises 1-12, (a) determine whether the set of vectors in R n is orthogonal, (b) if the set is orthogonal, then determine whether it is also orthonormal, and (c) determine whether the set is a basis for R n . { ( 2 , − 4 ) , ( 2 , 1 ) }

Orthogonal and Orthonormal Sets In Exercises 1-12, (a) determine whether the set of vectors in R n is orthogonal, (b) if the set is orthogonal, then determine whether it is also orthonormal, and (c) determine whether the set is a basis for R n . { ( 2 , − 4 ) , ( 2 , 1 ) }

Solution Summary: The author explains that a set of vectors (2,-4), left (2,1 )recht is orthogonal, but not orthonormal.

Elementary Linear Algebra (MindTap Course List)

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN: 9781305658004

Author: Ron Larson

Publisher: Cengage Learning

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Textbook Question

Chapter 5.3, Problem 1E

Orthogonal and Orthonormal Sets In Exercises 1-12, (a) determine whether the set of vectors in R n is orthogonal, (b) if the set is orthogonal, then determine whether it is also orthonormal, and (c) determine whether the set is a basis for R n .

{ ( 2 , − 4 ) , ( 2 , 1 ) }

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Chapter 5.2, Problem 101E

Chapter 5.3, Problem 2E

Chapter 5 Solutions

Elementary Linear Algebra (MindTap Course List)

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In Exercises...Ch. 5.1 - Prob. 61E Ch. 5.1 - Prob. 62E Ch. 5.1 - Prob. 63E Ch. 5.1 - Verifying the Pythagorean Theorem In Exercises...Ch. 5.1 - Prob. 65E Ch. 5.1 - Prob. 66E Ch. 5.1 - Rework Exercise 23 using matrix multiplication....Ch. 5.1 - Rework Exercise 24 using matrix multiplication....Ch. 5.1 - Prob. 69E Ch. 5.1 - Prob. 70E Ch. 5.1 - Writing In Exercises 71 and 72, determine whether...Ch. 5.1 - Prob. 72E Ch. 5.1 - True or False?In Exercises 73 and 74, determine...Ch. 5.1 - Prob. 74E Ch. 5.1 - Prob. 75E Ch. 5.1 - Prob. 76E Ch. 5.1 - Orthogonal Vectors In Exercises 77 and 78, let...Ch. 5.1 - Orthogonal Vectors In Exercises 77 and 78, let...Ch. 5.1 - Prob. 79E Ch. 5.1 - Prob. 80E Ch. 5.1 - Prob. 81E Ch. 5.1 - Prob. 82E Ch. 5.1 - Guided Proof Prove that if u is orthogonal to v...Ch. 5.1 - Prob. 84E Ch. 5.1 - Prob. 85E Ch. 5.1 - Proof Prove that u+v=u+v if and only if u and v...Ch. 5.1 - Proof Use the properties of matrix multiplication...Ch. 5.1 - Prob. 88E Ch. 5.1 - Writing Let x be a solution to mn homogeneous...Ch. 5.2 - Showing That a Function Is an Inner Product In...Ch. 5.2 - Showing That a Function Is an Inner Product In...Ch. 5.2 - Showing That a Function Is an Inner Product In...Ch. 5.2 - Showing That a Function Is an Inner Product In...Ch. 5.2 - Showing That a Function Is an Inner Product In...Ch. 5.2 - Showing That a Function Is an Inner Product In...Ch. 5.2 - Showing That a Function Is an Inner Product In...Ch. 5.2 - Showing That a Function Is an Inner ProductIn...Ch. 5.2 - Showing That a Function Is Not an Inner Product In...Ch. 5.2 - Showing That a Function Is Not an Inner Product In...Ch. 5.2 - Showing That a Function Is Not an Inner Product In...Ch. 5.2 - Showing That a Function Is Not an Inner Product In...Ch. 5.2 - Showing That a Function Is Not an Inner Product In...Ch. 5.2 - Showing That a Function Is Not an Inner Product In...Ch. 5.2 - Showing That a Function Is Not an Inner Product In...Ch. 5.2 - Prob. 16E Ch. 5.2 - Finding Inner Product, Length, and DistanceIn...Ch. 5.2 - Prob. 18E Ch. 5.2 - 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Prob. 40E Ch. 5.2 - Calculus In Exercises 39-42, use the functions f...Ch. 5.2 - Prob. 42E Ch. 5.2 - Finding the Angle Between Two Vectors In Exercises...Ch. 5.2 - Finding the Angle Between Two Vectors In Exercises...Ch. 5.2 - Finding the Angle Between Two Vectors In Exercises...Ch. 5.2 - Finding the Angle Between Two Vectors In Exercises...Ch. 5.2 - Finding the Angle Between Two Vectors In Exercises...Ch. 5.2 - Prob. 48E Ch. 5.2 - Finding the Angle Between Two Vectors In Exercises...Ch. 5.2 - Finding the Angle Between Two Vectors In Exercises...Ch. 5.2 - Prob. 51E Ch. 5.2 - Prob. 52E Ch. 5.2 - Prob. 53E Ch. 5.2 - Verifying Inequalities In Exercises 53-64, verify...Ch. 5.2 - Prob. 55E Ch. 5.2 - Prob. 56E Ch. 5.2 - Prob. 57E Ch. 5.2 - Prob. 58E Ch. 5.2 - Prob. 59E Ch. 5.2 - Verifying Inequalities In Exercises 53-64, verify...Ch. 5.2 - Verifying InequalitiesIn Exercises 53-64, verify a...Ch. 5.2 - Prob. 62E Ch. 5.2 - Prob. 63E Ch. 5.2 - Prob. 64E Ch. 5.2 - Calculus In Exercises 65-68, show that f and g are...Ch. 5.2 - Prob. 66E Ch. 5.2 - Calculus In Exercises 65-68, show that f and g are...Ch. 5.2 - Prob. 68E Ch. 5.2 - Prob. 69E Ch. 5.2 - Finding and Graphing Orthogonal Projections in R2...Ch. 5.2 - Prob. 71E Ch. 5.2 - Prob. 72E Ch. 5.2 - Prob. 73E Ch. 5.2 - Finding Orthogonal Projections In Exercises 7376,...Ch. 5.2 - Finding Orthogonal Projections In Exercises 7376,...Ch. 5.2 - Prob. 76E Ch. 5.2 - Prob. 77E Ch. 5.2 - Calculus In Exercises 77-84, find the orthogonal...Ch. 5.2 - Calculus In Exercises 77-84, find the orthogonal...Ch. 5.2 - Prob. 80E Ch. 5.2 - Prob. 81E Ch. 5.2 - Prob. 82E Ch. 5.2 - Prob. 83E Ch. 5.2 - Prob. 84E Ch. 5.2 - True or false?In Exercises 85 and 86, determine...Ch. 5.2 - Prob. 86E Ch. 5.2 - Prob. 87E Ch. 5.2 - Prob. 88E Ch. 5.2 - Prob. 89E Ch. 5.2 - Proof Let u and v be a nonzero vectors in an inner...Ch. 5.2 - Prob. 91E Ch. 5.2 - Prob. 92E Ch. 5.2 - Prob. 93E Ch. 5.2 - Prob. 94E Ch. 5.2 - Guided proofLet u,v be the Euclidean inner product...Ch. 5.2 - CAPSTONE (a) Explain how to determine whether a...Ch. 5.2 - 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Prob. 20E Ch. 5.3 - Finding a Coordinate Matrix In Exercises 19-24,...Ch. 5.3 - Finding a Coordinate Matrix In Exercises 19-24,...Ch. 5.3 - Prob. 23E Ch. 5.3 - Finding a Coordinate Matrix In Exercises 19-24,...Ch. 5.3 - Applying the Gram-Schmidt Process In Exercises...Ch. 5.3 - Prob. 26E Ch. 5.3 - Prob. 27E Ch. 5.3 - Prob. 28E Ch. 5.3 - Prob. 29E Ch. 5.3 - Prob. 30E Ch. 5.3 - Prob. 31E Ch. 5.3 - Prob. 32E Ch. 5.3 - Prob. 33E Ch. 5.3 - Prob. 34E Ch. 5.3 - Prob. 35E Ch. 5.3 - Prob. 36E Ch. 5.3 - Prob. 37E Ch. 5.3 - Prob. 38E Ch. 5.3 - Applying the Gram-Schmidt Process In Exercises...Ch. 5.3 - Prob. 40E Ch. 5.3 - Use the inner product u,v=2u1v1+u2v2 in R2 and...Ch. 5.3 - WritingExplain why the result of Exercise 41 is...Ch. 5.3 - Calculus In Exercises 43-48, let B={1,x,x2} be a...Ch. 5.3 - Prob. 44E Ch. 5.3 - Prob. 45E Ch. 5.3 - Prob. 46E Ch. 5.3 - Prob. 47E Ch. 5.3 - Calculus In Exercises 43-48, let B={1,x,x2} be a...Ch. 5.3 - Prob. 49E Ch. 5.3 - Prob. 50E Ch. 5.3 - Applying the Alternative Form of the Gram-Schmidt...Ch. 5.3 - Prob. 52E Ch. 5.3 - Applying the Alternative Form of the Gram-Schmidt...Ch. 5.3 - Prob. 54E Ch. 5.3 - Prob. 55E Ch. 5.3 - True or False? In Exercises 55 and 56, determine...Ch. 5.3 - Orthonormal Sets in P2In Exercises 57-62, let...Ch. 5.3 - Prob. 58E Ch. 5.3 - Prob. 59E Ch. 5.3 - Prob. 60E Ch. 5.3 - Orthonormal Sets in P2In Exercises 57-62, let...Ch. 5.3 - Orthonormal Sets in P2In Exercises 57-62, let...Ch. 5.3 - Prob. 63E Ch. 5.3 - Guided Proof Prove that if w is orthogonal to each...Ch. 5.3 - Prob. 65E Ch. 5.3 - Prob. 66E Ch. 5.3 - Prob. 67E Ch. 5.3 - Prob. 68E Ch. 5.3 - Prob. 69E Ch. 5.3 - Prob. 70E Ch. 5.3 - Prob. 71E Ch. 5.4 - Least Squares Regression LineIn Exercises 1-4,...Ch. 5.4 - Prob. 2E Ch. 5.4 - Prob. 3E Ch. 5.4 - Prob. 4E Ch. 5.4 - Prob. 5E Ch. 5.4 - Prob. 6E Ch. 5.4 - Prob. 7E Ch. 5.4 - Prob. 8E Ch. 5.4 - Prob. 9E Ch. 5.4 - Prob. 10E Ch. 5.4 - Prob. 11E Ch. 5.4 - Prob. 12E Ch. 5.4 - Prob. 13E Ch. 5.4 - Prob. 14E Ch. 5.4 - Prob. 15E Ch. 5.4 - Prob. 16E Ch. 5.4 - Prob. 17E Ch. 5.4 - Prob. 18E Ch. 5.4 - Prob. 19E Ch. 5.4 - Projection Onto a Subspace In Exercises 17-20,...Ch. 5.4 - Fundamental Subspaces In Exercises 21-24, find...Ch. 5.4 - Prob. 22E Ch. 5.4 - Prob. 23E Ch. 5.4 - Prob. 24E Ch. 5.4 - Prob. 25E Ch. 5.4 - Prob. 26E Ch. 5.4 - Finding the Least Squares Solutions In Exercises...Ch. 5.4 - Finding the Least Squares Solution In Exercises...Ch. 5.4 - Prob. 29E Ch. 5.4 - Prob. 30E Ch. 5.4 - Prob. 31E Ch. 5.4 - Prob. 32E Ch. 5.4 - Prob. 33E Ch. 5.4 - Prob. 34E Ch. 5.4 - Prob. 35E Ch. 5.4 - Prob. 36E Ch. 5.4 - Prob. 37E Ch. 5.4 - Prob. 38E Ch. 5.4 - Prob. 39E Ch. 5.4 - Prob. 40E Ch. 5.4 - Prob. 41E Ch. 5.4 - Prob. 42E Ch. 5.4 - True or false? In Exercises 43and 44, determine...Ch. 5.4 - True or false? In Exercises 43 and 44, determine...Ch. 5.4 - Proof Prove that if S1 and S2 are orthogonal...Ch. 5.4 - Prob. 46E Ch. 5.4 - Prob. 47E Ch. 5.4 - Prob. 48E Ch. 5.5 - Finding the Cross Product In Exercises 1-6, find...Ch. 5.5 - Prob. 2E Ch. 5.5 - Prob. 3E Ch. 5.5 - Prob. 4E Ch. 5.5 - Prob. 5E Ch. 5.5 - Prob. 6E Ch. 5.5 - Prob. 7E Ch. 5.5 - Prob. 8E Ch. 5.5 - Prob. 9E Ch. 5.5 - Prob. 10E Ch. 5.5 - Prob. 11E Ch. 5.5 - Prob. 12E Ch. 5.5 - Prob. 13E Ch. 5.5 - Prob. 14E Ch. 5.5 - Prob. 15E Ch. 5.5 - Prob. 16E Ch. 5.5 - Prob. 17E 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 - Prob. 23E Ch. 5.5 - Prob. 24E Ch. 5.5 - Prob. 25E Ch. 5.5 - Prob. 26E Ch. 5.5 - Prob. 27E Ch. 5.5 - Prob. 28E Ch. 5.5 - Prob. 29E Ch. 5.5 - Prob. 30E Ch. 5.5 - Prob. 31E Ch. 5.5 - Prob. 32E Ch. 5.5 - Prob. 33E Ch. 5.5 - Prob. 34E Ch. 5.5 - Prob. 35E Ch. 5.5 - Prob. 36E Ch. 5.5 - Prob. 37E Ch. 5.5 - Prob. 38E Ch. 5.5 - Prob. 39E Ch. 5.5 - Prob. 40E Ch. 5.5 - Prob. 41E Ch. 5.5 - Prob. 42E Ch. 5.5 - Prob. 43E Ch. 5.5 - Prob. 44E Ch. 5.5 - Prob. 45E Ch. 5.5 - Finding the Area of a Parallelogram In Exercises...Ch. 5.5 - Prob. 47E Ch. 5.5 - Prob. 48E Ch. 5.5 - Finding the Area of a Triangle In Exercises 49 and...Ch. 5.5 - Prob. 50E Ch. 5.5 - Prob. 51E Ch. 5.5 - Prob. 52E Ch. 5.5 - Prob. 53E Ch. 5.5 - Prob. 54E Ch. 5.5 - Prob. 55E Ch. 5.5 - Prob. 56E Ch. 5.5 - Prob. 57E Ch. 5.5 - Prob. 58E Ch. 5.5 - Prob. 59E Ch. 5.5 - Prob. 60E Ch. 5.5 - Prob. 61E Ch. 5.5 - Prob. 62E Ch. 5.5 - Prob. 63E Ch. 5.5 - Prob. 64E Ch. 5.5 - Prob. 65E Ch. 5.5 - Prob. 66E Ch. 5.5 - Prob. 67E Ch. 5.5 - Prob. 68E Ch. 5.5 - Prob. 69E Ch. 5.5 - Prob. 70E Ch. 5.5 - Prob. 71E Ch. 5.5 - Prob. 72E Ch. 5.5 - Prob. 73E Ch. 5.5 - Prob. 74E Ch. 5.5 - Finding a Least Squares Approximation In Exercises...Ch. 5.5 - Prob. 76E Ch. 5.5 - Prob. 77E Ch. 5.5 - Prob. 78E Ch. 5.5 - Prob. 79E Ch. 5.5 - Prob. 80E Ch. 5.5 - Prob. 81E Ch. 5.5 - Prob. 82E Ch. 5.5 - Prob. 83E Ch. 5.5 - Prob. 84E Ch. 5.5 - Prob. 85E Ch. 5.5 - Prob. 86E Ch. 5.5 - Prob. 87E Ch. 5.5 - Prob. 88E Ch. 5.5 - Prob. 89E Ch. 5.5 - Prob. 90E Ch. 5.5 - Prob. 91E Ch. 5.5 - Prob. 92E Ch. 5.5 - Use your schools library, the Internet, or some...Ch. 5.CR - Finding Lengths, Dot Product, and Distance In...Ch. 5.CR - Finding Lengths, Dot Product, and Distance In...Ch. 5.CR - Prob. 3CR Ch. 5.CR - Prob. 4CR Ch. 5.CR - Finding Lengths, Dot Product, and Distance In...Ch. 5.CR - Finding Lengths, Dot Product, and Distance In...Ch. 5.CR - Finding Lengths, Dot Product, and Distance In...Ch. 5.CR - Finding Lengths, Dot Product, and Distance In...Ch. 5.CR - Prob. 9CR Ch. 5.CR - Prob. 10CR Ch. 5.CR - Prob. 11CR Ch. 5.CR - Prob. 12CR Ch. 5.CR - Prob. 13CR Ch. 5.CR - Prob. 14CR Ch. 5.CR - Prob. 15CR Ch. 5.CR - Prob. 16CR Ch. 5.CR - Prob. 17CR Ch. 5.CR - Prob. 18CR Ch. 5.CR - Finding the Angle Between Two VectorsIn Exercises...Ch. 5.CR - Finding the Angle Between Two Vectors In Exercises...Ch. 5.CR - Prob. 21CR Ch. 5.CR - Prob. 22CR Ch. 5.CR - Prob. 23CR Ch. 5.CR - 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Find an orthonormal basis for the subspace of...Ch. 5.CR - Find an orthonormal basis for the solution space...Ch. 5.CR - Prob. 49CR Ch. 5.CR - Prob. 50CR Ch. 5.CR - Prob. 51CR Ch. 5.CR - Prob. 52CR Ch. 5.CR - Prob. 53CR Ch. 5.CR - Let V be an two dimensional subspace of R4 spanned...Ch. 5.CR - Prob. 55CR Ch. 5.CR - Prob. 56CR Ch. 5.CR - Prob. 57CR Ch. 5.CR - Prob. 58CR Ch. 5.CR - Prob. 59CR Ch. 5.CR - Find the projection of the vector v=[102]T onto...Ch. 5.CR - Find the bases for the four fundamental subspaces...Ch. 5.CR - Prob. 62CR Ch. 5.CR - Prob. 63CR Ch. 5.CR - Prob. 64CR Ch. 5.CR - Finding the Cross Product In Exercises 65-68, find...Ch. 5.CR - Finding the Cross Product In Exercises 65-68, find...Ch. 5.CR - Prob. 67CR Ch. 5.CR - Finding the Cross Product In Exercises 65-68, find...Ch. 5.CR - Prob. 69CR Ch. 5.CR - Prob. 70CR Ch. 5.CR - Finding the Volume of a ParallelepipedIn Exercises...Ch. 5.CR - Prob. 72CR Ch. 5.CR - Prob. 73CR Ch. 5.CR - Prob. 74CR Ch. 5.CR - Finding a Least Approximation In Exercises 75-78,...Ch. 5.CR - Finding a Least Approximation In Exercises 75-78,...Ch. 5.CR - Prob. 77CR Ch. 5.CR - Finding a Least Approximation In Exercises 75-78,...Ch. 5.CR - Finding a Least Squares Approximation In Exercises...Ch. 5.CR - Finding a Least Squares Approximation In Exercises...Ch. 5.CR - Prob. 81CR Ch. 5.CR - Prob. 82CR Ch. 5.CR - Prob. 83CR Ch. 5.CR - Prob. 84CR Ch. 5.CM - Prob. 1CM Ch. 5.CM - Take this test to review the material in Chapters...Ch. 5.CM - Take this test to review the material in Chapters...Ch. 5.CM - Use a software program or a graphing utility to...Ch. 5.CM - Take this test to review the material in Chapters...Ch. 5.CM - Prob. 6CM Ch. 5.CM - Prob. 7CM Ch. 5.CM - Take this test to review the material in Chapters...Ch. 5.CM - Take this test to review the material in Chapters...Ch. 5.CM - Prob. 10CM Ch. 5.CM - Prob. 11CM Ch. 5.CM - Prob. 12CM Ch. 5.CM - Prob. 13CM Ch. 5.CM - Prob. 14CM Ch. 5.CM - Prob. 15CM Ch. 5.CM - Prob. 16CM Ch. 5.CM - Prob. 17CM Ch. 5.CM - Prob. 18CM Ch. 5.CM - Prob. 19CM Ch. 5.CM - Prob. 20CM Ch. 5.CM - Prob. 21CM Ch. 5.CM - The two matrices A and B are row-equivalent....Ch. 5.CM - Prob. 23CM Ch. 5.CM - Prob. 24CM

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