Linear Algebra and Its Applications (5th Edition)
5th Edition
ISBN: 9780321982384
Author: David C. Lay, Steven R. Lay, Judi J. McDonald
Publisher: PEARSON
expand_more
expand_more
format_list_bulleted
Concept explainers
Textbook Question
Chapter 6.7, Problem 16E
Use the inner product axioms and other results of this section to verify the statements in Exercises 15-18.
16. If {u, v} is an orthonormal set in V, then
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
Use the inner product axioms and other results of this section to
verify the statements in Exercises 15-18.
16. If {u, v} is an orthonormal set in V, then ||uv|| = √2.
-
If A+ iB is Hermitian (A and Bare real) show that [ AB -BA] is symmetric.
In cartesian product, prove that
(A-B)XC=(AXC)-(BXC)
A,B,C are sets
Chapter 6 Solutions
Linear Algebra and Its Applications (5th Edition)
Ch. 6.1 - Let a = [21] and b = [31]. Compute abaa and...Ch. 6.1 - Let c = [4/312/3] and d = [561]. a. Find a unit...Ch. 6.1 - Let W be a subspace of Rn. Exercise 30 establishes...Ch. 6.1 - Compute the quantities in Exercises 18 using the...Ch. 6.1 - Compute the quantities in Exercises 18 using the...Ch. 6.1 - Compute the quantities in Exercises 18 using the...Ch. 6.1 - Compute the quantities in Exercises 18 using the...Ch. 6.1 - Compute the quantities in Exercises 18 using the...Ch. 6.1 - Compute the quantities in Exercises 18 using the...Ch. 6.1 - Compute the quantities in Exercises 18 using the...
Ch. 6.1 - Compute the quantities in Exercises 18 using the...Ch. 6.1 - In Exercises 912, find a unit vector in the...Ch. 6.1 - In Exercises 912, find a unit vector in the...Ch. 6.1 - In Exercises 912, find a unit vector in the...Ch. 6.1 - Prob. 12ECh. 6.1 - Find the distance between x = [103] and y = [15].Ch. 6.1 - Find the distance between u = [052] and z = [418].Ch. 6.1 - Determine which pairs of vectors in Exercises 1518...Ch. 6.1 - Determine which pairs of vectors in Exercises 1518...Ch. 6.1 - Determine which pairs of vectors in Exercises 1518...Ch. 6.1 - Determine which pairs of vectors in Exercises 1518...Ch. 6.1 - In Exercises 19 and 20, all vectors are in n. Mark...Ch. 6.1 - In Exercises 19 and 20, all vectors are in n. Mark...Ch. 6.1 - Use the transpose definition of the inner product...Ch. 6.1 - Prob. 22ECh. 6.1 - Let u = [251] and v = [746]. Compute and compare...Ch. 6.1 - Verify the parallelogram law for vectors u and v...Ch. 6.1 - Let v = [ab] Describe the set H of vectors [xy]...Ch. 6.1 - Let u = [567], and let W be the set of all x in 3...Ch. 6.1 - Suppose a vector y is orthogonal to vectors u and...Ch. 6.1 - Suppose y is orthogonal to u and v. Show that y is...Ch. 6.1 - Let W = Span {v1,,vp}. Show that if x is...Ch. 6.1 - Let W be a subspace of n, and let W be the set of...Ch. 6.1 - Show that if x is in both W and W, then x = 0.Ch. 6.2 - Let u1= [1/52/5] and u2= [2/51/5]. Show that {u1....Ch. 6.2 - Let y and L be as in Example 3 and Figure 3....Ch. 6.2 - Let U and x be as in Example 6. and let y = [326]....Ch. 6.2 - Let U be an n n matrix with orthonormal columns....Ch. 6.2 - In Exercises 16, determine which sets of vectors...Ch. 6.2 - In Exercises 16, determine which sets of vectors...Ch. 6.2 - In Exercises 16, determine which sets of vectors...Ch. 6.2 - In Exercises 16, determine which sets of vectors...Ch. 6.2 - In Exercises 16, determine which sets of vectors...Ch. 6.2 - In Exercises 16, determine which sets of vectors...Ch. 6.2 - In Exercises 710, show that {u1, u2} or {u1, u2,...Ch. 6.2 - In Exercises 710, show that {u1, u2} or {u1, u2,...Ch. 6.2 - In Exercises 710, show that {u1, u2} or {u1, u2,...Ch. 6.2 - In Exercises 710, show that {u1, u2} or {u1, u2,...Ch. 6.2 - Compute the orthogonal projection of [17] onto the...Ch. 6.2 - Compute the orthogonal projection of [11] onto the...Ch. 6.2 - Let y = [23] and u = [47] Write y as the sum of...Ch. 6.2 - Let y = [26] and u = [71] Write y as the sum of a...Ch. 6.2 - Let y = [31] and u = [86] Compute the distance...Ch. 6.2 - Let y = [39] and u = [12] Compute the distance...Ch. 6.2 - In Exercises 1722, determine which sets of vectors...Ch. 6.2 - In Exercises 1722, determine which sets of vectors...Ch. 6.2 - In Exercises 1722, determine which sets of vectors...Ch. 6.2 - In Exercises 1722, determine which sets of vectors...Ch. 6.2 - In Exercises 1722, determine which sets of vectors...Ch. 6.2 - In Exercises 1722, determine which sets of vectors...Ch. 6.2 - In Exercises 23 and 24, all vectors are in n. Mark...Ch. 6.2 - In Exercises 23 and 24, all vectors are in n. Mark...Ch. 6.2 - Prove Theorem 7. [Hint: For (a), compute |Ux||2,...Ch. 6.2 - Suppose W is a sub space of n spanned by n nonzero...Ch. 6.2 - Let U be a square matrix with orthonormal columns....Ch. 6.2 - Let U be an n n orthogonal matrix. Show that the...Ch. 6.2 - Let U and V be n n orthogonal matrices. Explain...Ch. 6.2 - Let U be an orthogonal matrix, and construct V by...Ch. 6.2 - Show that the orthogonal projection of a vector y...Ch. 6.2 - Let {v1, v2} be an orthogonal set of nonzero...Ch. 6.2 - Prob. 33ECh. 6.2 - Given u 0 in n, let L = Span{u}. For y in n, the...Ch. 6.3 - Let u1 = [714], u2 = [112], x = [916], and W =...Ch. 6.3 - Let W be a subspace of n. Let x and y be vectors...Ch. 6.3 - In Exercises 1 and 2, you may assume that {u1,,...Ch. 6.3 - u1 = [1211], u2 = [2111], u3 = [1121], u4 =...Ch. 6.3 - In Exercises 36, verify that {u1, u2} is an...Ch. 6.3 - In Exercises 36, verify that {u1, u2} is an...Ch. 6.3 - In Exercises 36, verify that {u1, u2} is an...Ch. 6.3 - In Exercises 36, verify that {u1, u2} is an...Ch. 6.3 - In Exercises 710, let W be the subspace spanned by...Ch. 6.3 - In Exercises 710, let W be the subspace spanned by...Ch. 6.3 - In Exercises 710, let W be the subspace spanned by...Ch. 6.3 - In Exercises 710, let W be the subspace spanned by...Ch. 6.3 - In Exercises 11 and 12, find the closest point to...Ch. 6.3 - In Exercises 11 and 12, find the closest point to...Ch. 6.3 - In Exercises 13 and 14, find the best...Ch. 6.3 - In Exercises 13 and 14, find the best...Ch. 6.3 - Let y = [595], u1 = [351], u2 = [321]. Find die...Ch. 6.3 - Let y, v1, and v2 be as in Exercise 12. Find the...Ch. 6.3 - Let y = [481], u1 = [2/31/32/3], u2 = [2/32/31/3],...Ch. 6.3 - Let y = [79], u1 = [1/103/10], and W = Span {u1}....Ch. 6.3 - Let u1 = [112], u2 = [512], and u3 = [001].Note...Ch. 6.3 - Let u1 and u2 be as in Exercise 19, and let u4 =...Ch. 6.3 - In Exercises 21 and 22, all vectors and subspaces...Ch. 6.3 - In Exercises 21 and 22, all vectors and subspaces...Ch. 6.3 - Let A be an m m matrix. Prove that every vector x...Ch. 6.3 - Let W be a subspace of n with an orthogonal basis...Ch. 6.4 - Let W = Span {x1, x2}, where x1 = [111] and x2 =...Ch. 6.4 - Suppose A = QR, where Q is an m n matrix with...Ch. 6.4 - In Exercises 1-6, the given set is a basis for a...Ch. 6.4 - In Exercises 1-6, the given set is a basis for a...Ch. 6.4 - In Exercises 1-6, the given set is a basis for a...Ch. 6.4 - In Exercises 1-6, the given set is a basis for a...Ch. 6.4 - In Exercises 1-6, the given set is a basis for a...Ch. 6.4 - In Exercises 1-6, the given set is a basis for a...Ch. 6.4 - Find an orthonormal basis of the subspace spanned...Ch. 6.4 - Find an orthonormal basis of the subspace spanned...Ch. 6.4 - Find an orthogonal basis for the column space of...Ch. 6.4 - Find an orthogonal basis for the column space of...Ch. 6.4 - Find an orthogonal basis for the column space of...Ch. 6.4 - Find an orthogonal basis for the column space of...Ch. 6.4 - In Exercises 13 and 14, the columns of Q were...Ch. 6.4 - In Exercises 13 and 14, the columns of Q were...Ch. 6.4 - Find a QR factorization of the matrix in Exercise...Ch. 6.4 - Find a QR factorization of the matrix in Exercise...Ch. 6.4 - In Exercises 17 and 18, all vectors and subspaces...Ch. 6.4 - In Exercises 17 and 18, all vectors and subspaces...Ch. 6.4 - Suppose A = QR, where Q is m n and R is n n....Ch. 6.4 - Suppose A = QR, where R is an invertible matrix....Ch. 6.4 - Given A = QR as in Theorem 12, describe how to...Ch. 6.4 - Let u1, , up be an orthogonal basis for a subspace...Ch. 6.4 - Suppose A = QR is a QR factorization of an m n...Ch. 6.4 - [M] Use the Gram-Schmidt process as in Example 2...Ch. 6.4 - [M] Use the method in this section to produce a QR...Ch. 6.5 - Let A = [133151172] and b = [535]. Find a...Ch. 6.5 - What can you say about the least-squares solution...Ch. 6.5 - In Exercises 1-4, find a least-squares solution of...Ch. 6.5 - In Exercises 1-4, find a least-squares solution of...Ch. 6.5 - In Exercises 1-4, find a least-squares solution of...Ch. 6.5 - In Exercises 1-4, find a least-squares solution of...Ch. 6.5 - In Exercises 5 and 6, describe all least-squares...Ch. 6.5 - In Exercises 5 and 6, describe all least-squares...Ch. 6.5 - Compute the least-squares error associated with...Ch. 6.5 - Compute the least-squares error associated with...Ch. 6.5 - In Exercises 9-12, find (a) the orthogonal...Ch. 6.5 - In Exercises 9-12, find (a) the orthogonal...Ch. 6.5 - In Exercises 9-12, find (a) the orthogonal...Ch. 6.5 - In Exercises 9-12, find (a) the orthogonal...Ch. 6.5 - Let A = [342134], b = [1195], u = [51], and v =...Ch. 6.5 - Let A = [213432], b = [544], u = [45], and v =...Ch. 6.5 - In Exercises 15 and 16, use the factorization A =...Ch. 6.5 - In Exercises 15 and 16, use the factorization A =...Ch. 6.5 - In Exercises 17 and 18, A is an m n matrix and b...Ch. 6.5 - a. If b is in the column space of A, then every...Ch. 6.5 - Let A be an m n matrix. Use the steps below to...Ch. 6.5 - Let A be an m n matrix such that ATA is...Ch. 6.5 - Let A be an m n matrix whose columns are linearly...Ch. 6.5 - Use Exercise 19 to show that rank ATA = rank A....Ch. 6.5 - Suppose A is m n with linearly independent...Ch. 6.5 - Find a formula for the least-squares solution of...Ch. 6.5 - Describe all least-squares solutions of the system...Ch. 6.6 - When the monthly sales of a product are subject to...Ch. 6.6 - In Exercises 1-4, find the equation y = 0 + 1x of...Ch. 6.6 - In Exercises 1-4, find the equation y = 0 + 1x of...Ch. 6.6 - In Exercises 1-4, find the equation y = 0 + 1x of...Ch. 6.6 - In Exercises 1-4, find the equation y = 0 + 1x of...Ch. 6.6 - Let X be the design matrix used to find the...Ch. 6.6 - Let X be the design matrix in Example 2...Ch. 6.6 - A certain experiment produces the data (1, 7.9),...Ch. 6.6 - Let x=1n(x1++xn) and y=1n(y1++yn). Show that the...Ch. 6.6 - Derive the normal equations (7) from the matrix...Ch. 6.6 - Use a matrix inverse to solve the system of...Ch. 6.6 - a. Rewrite the data in Example 1 with new...Ch. 6.6 - Suppose the x-coordinates of the data (x1, y1), ,...Ch. 6.6 - Exercises 19 and 20 involve a design matrix X with...Ch. 6.6 - Show that X2=TXTy. [Hint: Rewrite the left side...Ch. 6.7 - Use the inner product axioms to verify the...Ch. 6.7 - Use the inner product axioms to verify the...Ch. 6.7 - Let 2 have the inner product of Example 1, and let...Ch. 6.7 - Let 2 have the inner product of Example 1. Show...Ch. 6.7 - Exercises 3-8 refer to 2 with the inner product...Ch. 6.7 - Exercises 3-8 refer to 2 with the inner product...Ch. 6.7 - Exercises 3-8 refer to 2 with the inner product...Ch. 6.7 - Exercises 3-8 refer to 2 with the inner product...Ch. 6.7 - Exercises 3-8 refer to 2 with the inner product...Ch. 6.7 - Exercises 3-8 refer to 2 with the inner product...Ch. 6.7 - Let 3 have the inner product given by evaluation...Ch. 6.7 - Let 3 have the inner product as in Exercise 9,...Ch. 6.7 - Let p0, p1, and p2 be the orthogonal polynomials...Ch. 6.7 - Find a polynomial p3 such that {p0, p1, p2, p3}...Ch. 6.7 - Let A be any invertible n n matrix. Show that for...Ch. 6.7 - Let T be a one-to-one linear transformation from a...Ch. 6.7 - Use the inner product axioms and other results of...Ch. 6.7 - Use the inner product axioms and other results of...Ch. 6.7 - Use the inner product axioms and other results of...Ch. 6.7 - Use the inner product axioms and other results of...Ch. 6.7 - Given a 0 and b 0, let u=[ab] and v=[ba]. Use...Ch. 6.7 - Let u=[ab] and v=[11]. Use the Cauchy-Schwarz...Ch. 6.7 - Exercises 21-24 refer to V = C[0, 1], with the...Ch. 6.7 - Exercises 21-24 refer to V = C[0, 1], with the...Ch. 6.7 - Compute f for f in Exercise 21. Exercises 21-24...Ch. 6.7 - Compute g for g in Exercise 22. Exercises 21-24...Ch. 6.7 - Let V be the space C[1, 1] with the inner product...Ch. 6.7 - Let V be the space C[2, 2] with the inner product...Ch. 6.8 - Let q1(t) = 1, q2(t) = t, and q3(t) = 3t2 4....Ch. 6.8 - Find the first-order and third-order Fourier...Ch. 6.8 - Find the least-squares line y = 0 + 1x that best...Ch. 6.8 - Suppose 5 out of 25 data points in a weighted...Ch. 6.8 - Fit a cubic trend function to the data in Example...Ch. 6.8 - To make a trend analysis of six evenly spaced data...Ch. 6.8 - In Exercises 5-14, the space is C[0, 2] with the...Ch. 6.8 - In Exercises 5-14, the space is C[0, 2] with the...Ch. 6.8 - Prob. 7ECh. 6.8 - In Exercises 5-14, the space is C[0, 2] with the...Ch. 6.8 - In Exercises 5-14, the space is C[0, 2] with the...Ch. 6.8 - In Exercises 5-14, the space is C[0, 2] with the...Ch. 6.8 - In Exercises 5-14, the space is C[0, 2] with the...Ch. 6.8 - In Exercises 5-14, the space is C[0, 2] with the...Ch. 6.8 - In Exercises 5-14, the space is C[0, 2] with the...Ch. 6.8 - In Exercises 5-14, the space is C[0, 2] with the...Ch. 6.8 - [M] Refer to the data in Exercise 13 in Section...Ch. 6.8 - [M] Let f4 and f5 be the fourth-order and...Ch. 6 - Prob. 1SECh. 6 - Prob. 2SECh. 6 - Let {v1, , vp} be an orthonormal set in n. Verify...Ch. 6 - Let U be an n n orthogonal matrix. Show that if...Ch. 6 - Show that if an n n matrix U satisfies (Ux) (Uy)...Ch. 6 - Show that if U is an orthogonal matrix, then any...Ch. 6 - A Householder matrix, or an elementary reflector,...Ch. 6 - Let T: n n be a linear transformation that...Ch. 6 - Let u and v be linearly independent vectors in n...Ch. 6 - Suppose the columns of A are linearly independent....Ch. 6 - If a, b, and c are distinct numbers, then the...Ch. 6 - Consider the problem of finding an eigenvalue of...Ch. 6 - Use the steps below to prove the following...Ch. 6 - Explain why an equation Ax = b has a solution if...Ch. 6 - Exercises 15 and 16 concern the (real) Schur...Ch. 6 - Let A be an n n matrix with n real eigenvalues,...
Additional Math Textbook Solutions
Find more solutions based on key concepts
1. How much money is Joe earning when he’s 30?
Pathways To Math Literacy (looseleaf)
True or False The quotient of two polynomial expressions is a rational expression, (p. A35)
Precalculus
23. A plant nursery sells two sizes of oak trees to landscapers. Large trees cost the nursery $120 from the gro...
College Algebra (Collegiate Math)
Find E(X) for each of the distributions given in Exercise 2.1-3.
Probability And Statistical Inference (10th Edition)
Evaluate the integrals in Exercises 1–46.
1.
University Calculus: Early Transcendentals (4th Edition)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.Similar questions
- we have seen thatarrow_forwardUse the inner product axioms and other results to verify the statement. Assume c is a scalar and u, v, and exist in V, a vector space with an inner product denoted (u,v). (u,cv) = c (u,v) Apply the inner product axioms to the expression on the left side of the given statement, (u,cv), to derive the right side c (u,v). Which inner product axiom should be applied first? O A. Ju|| = V(u,u) or ||u|| = (u,u) О в. (и,у) (v,u) O c. (cu,v) =c (u,v) OD. (u + v,w) = (u,w) + (v,w) O E. (u,u) 20 and (u,u) = 0 if and only if u = 0 Apply the axiom from the previous step. (u,cv) = Which inr be applied next? O A. (I ind only if u = 0 c(v,u) О В. ( (cv,u) O C. 1,u) O D. (I O E. ( c{u,v) Apply the step to the previous result. (u,c) + (u,v) (u.cv) = (u,v)c Which inr nition shows that the result from the previous step is equivalent to c(u,v) ?arrow_forward4) which of the following is a basis for the null space of -88 88 -8E 888 b)arrow_forward
- The number of elements of the set M $M=\{\hat{x}^2$\ \hspace{0.2} $|\hat{x}\in \mathbb{Z}_{8}\}$\ is: a) 3; b) 2; c) 4; d) 5 wich answer is the corect one? mention that I need a complete solution of the exercise with steps and details. There is no additional details for this question! Thank you in advance. please look at the picture uploaded in the file for clarity.arrow_forwardLet (x, y) stands for a complex number z EC where x = Re(z) and y = Im(z). Now, let us define a new binary operation * on C by (X1, Y₁) *k (X2, Y2) def (X1X2 + Y1Y2, X2Y1 — X1Y2 ). - (a) Determine if this binary operation * is commutative on C. Justify your answer. (b) Determine if this binary operation is associative on C. Justify your answer. (c) For (x₁, y₁) # (0, 0),there is an unique (x2, y2) such that (x₁, y₁) *k (X2, Y2) = (1, 0). Write (x2, 92₂) in terms of (x₁, y₁).arrow_forwardA https://docs.google.com/forms/d/e/1FAlpQLSeUGYc-JH5nqoxlHtmLUWA46QI 150% Search Let a E R which expressions are correct? and give the justification why they are part of the Borel o-algebra, B(R) U (-) («) - U|--) {a} {a} nEN nEN Q = U{a;}, a; is rational U {a;}, a, is irrational iEN iEN searcharrow_forward
- Let G= (i. e G={1,a,b,a.b}). Find all the conjugate classes of (G, .).arrow_forwardLet U = {1,2,3,4,5), A={1,3,5), B = {1,3,4), and C = {2,3,5). (a) Find A U (BUC). (b) Find (AUB) U C. (c) State a conjecture. Use the results in parts (a) and (b) to answer this part. (a) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. AU (BUC)=1 OB. AU (BUC) = 0 (Use a comma to separate answers as needed.)arrow_forwardFind a basis for span(l - x, x - x2, 1 - x2, 1 - 2x + x2) in P 2arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningElements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Elementary Geometry For College Students, 7eGeometryISBN:9781337614085Author:Alexander, Daniel C.; Koeberlein, Geralyn M.Publisher:Cengage,
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:9781285463247
Author:David Poole
Publisher:Cengage Learning
Elements Of Modern Algebra
Algebra
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Cengage Learning,
Elementary Geometry For College Students, 7e
Geometry
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Cengage,
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:9781305658004
Author:Ron Larson
Publisher:Cengage Learning
Propositional Logic, Propositional Variables & Compound Propositions; Author: Neso Academy;https://www.youtube.com/watch?v=Ib5njCwNMdk;License: Standard YouTube License, CC-BY
Propositional Logic - Discrete math; Author: Charles Edeki - Math Computer Science Programming;https://www.youtube.com/watch?v=rL_8y2v1Guw;License: Standard YouTube License, CC-BY
DM-12-Propositional Logic-Basics; Author: GATEBOOK VIDEO LECTURES;https://www.youtube.com/watch?v=pzUBrJLIESU;License: Standard Youtube License
Lecture 1 - Propositional Logic; Author: nptelhrd;https://www.youtube.com/watch?v=xlUFkMKSB3Y;License: Standard YouTube License, CC-BY
MFCS unit-1 || Part:1 || JNTU || Well formed formula || propositional calculus || truth tables; Author: Learn with Smily;https://www.youtube.com/watch?v=XV15Q4mCcHc;License: Standard YouTube License, CC-BY