Linear Algebra: A Modern Introduction
4th Edition
ISBN: 9781285463247
Author: David Poole
Publisher: Cengage Learning
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Chapter 3.2, Problem 21EQ
Prove the half of Theorem 3.3 (e) that was not proved in the text.
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You are given a plane Π in R3 defined by two vectors, p1 and p2, and a subspace W in R3 spanned by twovectors, w1 and w2. Your task is to project the plane Π onto the subspace W.First, answer the question of what the projection matrix is that projects onto the subspace W and how toapply it to find the desired projection. Second, approach the task in a different way by using the Gram-Schmidtmethod to find an orthonormal basis for subspace W, before then using the resulting basis vectors for theprojection. Last, compare the results obtained from both methods
Chapter 3 Solutions
Linear Algebra: A Modern Introduction
Ch. 3.1 - Let...Ch. 3.1 - Let
In Exercises 1-16, compute the indicated...Ch. 3.1 - Let...Ch. 3.1 - Let
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In Exercises 1-16, compute the indicated...Ch. 3.1 - Let
In Exercises 1-16, compute the indicated...Ch. 3.1 - Let...Ch. 3.1 - Let...Ch. 3.1 - Let...
Ch. 3.1 - Let
In Exercises 1-16, compute the indicated...Ch. 3.1 - Let
In Exercises 1-16, compute the indicated...Ch. 3.1 - Let...Ch. 3.1 - Let...Ch. 3.1 - Let...Ch. 3.1 - Let...Ch. 3.1 - Give an example of a nonzero 22 matrix A such that...Ch. 3.1 - Let A=[2613]. Find 22 matrices B and C such that...Ch. 3.1 - A factory manufactures three products (doohickies,...Ch. 3.1 - Referring to Exercise 19, suppose that the unit...Ch. 3.1 - In Exercises 21-22, write the given system of...Ch. 3.1 - In Exercises 21-22, write the given system of...Ch. 3.1 - In Exercises 23-28, let A=[102311201] and...Ch. 3.1 - In Exercises 23-28, let
and
24. Use the...Ch. 3.1 - In Exercises 23-28, let
and
25. Compute the...Ch. 3.1 - In Exercises 23-28, let A=[102311201] and...Ch. 3.1 - In Exercises 23-28, let
and
27. Use the...Ch. 3.1 - Prob. 28EQCh. 3.1 - In Exercises 29 and 30, assume that the product AB...Ch. 3.1 - Prob. 30EQCh. 3.1 -
In Exercises 31-34, compute AB by block...Ch. 3.1 - In Exercises 31-34, compute AB by block...Ch. 3.1 - In Exercises 31-34, compute AB by block...Ch. 3.1 - In Exercises 31-34, compute AB by block...Ch. 3.1 - Prob. 35EQCh. 3.1 - Let B=[12121212]. Find, with justification, B2015.Ch. 3.1 - Let A=[1101]. Find a formula for An(n1) and verify...Ch. 3.1 - 38. Let
(a) Show that
(b) Prove, by mathematical...Ch. 3.1 - In each of the following, find the 66matrixA=[aij]...Ch. 3.2 - In Exercises 1-4, solve the equation for X, given...Ch. 3.2 - In Exercises 1-4, solve the equation for X, given...Ch. 3.2 - In Exercises 1-4, solve the equation for X, given...Ch. 3.2 - In Exercises 1-4, solve the equation for X, given...Ch. 3.2 - In Exercises 5-8, write B as a linear combination...Ch. 3.2 - In Exercises 5-8, write B as a linear combination...Ch. 3.2 - In Exercises 5-8, write B as a linear combination...Ch. 3.2 - In Exercises 5-8, write B as a linear combination...Ch. 3.2 - In Exercises 9-12, find the general form of the...Ch. 3.2 - In Exercises 9-12, find the general form of the...Ch. 3.2 - In Exercises 9-12, find the general form of the...Ch. 3.2 - In Exercises 9-12, find the general form of the...Ch. 3.2 - In Exercises 13-16, determine whether the given...Ch. 3.2 - In Exercises 13-16, determine whether the given...Ch. 3.2 - In Exercises 13-16, determine whether the given...Ch. 3.2 - In Exercises 13-16, determine whether the given...Ch. 3.2 - 17. Prove Theorem 3.2(a) -(d).Ch. 3.2 - Prove Theorem 3.2 (e) (h).Ch. 3.2 - Prove Theorem 3.3(c).Ch. 3.2 - Prove Theorem 3.3(d).Ch. 3.2 - Prove the half of Theorem 3.3 (e) that was not...Ch. 3.2 - 22. Prove that, for square matrices A and B, AB =...Ch. 3.2 - In Exercises 23-25, if , find conditions on a, b,...Ch. 3.2 - In Exercises 23-25, if B=[abcd], find conditions...Ch. 3.2 - In Exercises 23-25, B=[abcd], find conditions on...Ch. 3.2 - 26. Find conditions on a, b, c, and d such that ...Ch. 3.2 - 27. Find conditions on a, b, c, and d such that ...Ch. 3.2 - Prove that if AB and BA are both defined, then AB...Ch. 3.2 - A square matrix is called upper triangular if all...Ch. 3.2 - 33. Using induction, prove that for all
.
Ch. 3.3 - In Exercises 1-10, find the inverse of the given...Ch. 3.3 - In Exercises 1-10, find the inverse of the given...Ch. 3.3 - In Exercises 1-10, find the inverse of the given...Ch. 3.3 - In Exercises 1-10, find the inverse of the given...Ch. 3.3 - In Exercises 5-8, write B as a linear combination...Ch. 3.3 - Prob. 6EQCh. 3.3 - In Exercises 1-10, find the inverse of the given...Ch. 3.3 - In Exercises 1-10, find the inverse of the given...Ch. 3.3 - In Exercises 1-10, find the inverse of the given...Ch. 3.3 - In Exercises 1-10, find the inverse of the given...Ch. 3.3 - In Exercises 11 and 12, solve the given system...Ch. 3.3 - In Exercises 11 and 12, solve the given system...Ch. 3.3 - Let A=[1226],b1=[35],b2=[12],andb3=[20]. Find A-1...Ch. 3.3 - In Exercises 20-23, solve the given matrix...Ch. 3.3 - In Exercises 20-23, solve the given matrix...Ch. 3.3 - In Exercises 20-23, solve the given matrix...Ch. 3.3 - In Exercises 20-23, solve the given matrix...Ch. 3.3 - In Exercises let
In each case, find an...Ch. 3.3 - Prob. 25EQCh. 3.3 - Prob. 26EQCh. 3.3 - Prob. 27EQCh. 3.3 - Prob. 28EQCh. 3.3 - Prob. 29EQCh. 3.3 - Prob. 30EQCh. 3.3 - Prob. 31EQCh. 3.3 - Prob. 32EQCh. 3.3 - In Exercises 31-38, find the inverse of the given...Ch. 3.3 - In Exercises 31-38, find the inverse of the given...Ch. 3.3 - In Exercises 31-38, find the inverse of the given...Ch. 3.3 - In Exercises 31-38, find the inverse of the given...Ch. 3.3 - Prob. 48EQCh. 3.3 - Prob. 49EQCh. 3.3 - In Exercises 48-63, use the Gauss-Jordan method to...Ch. 3.3 - Prob. 51EQCh. 3.3 - In Exercises 48-63, use the Gauss-Jordan method to...Ch. 3.3 - In Exercises 48-63, use the Gauss-Jordan method to...Ch. 3.3 - Prob. 54EQCh. 3.3 - Prob. 55EQCh. 3.3 - In Exercises 48-63, use the Gauss-Jordan method to...Ch. 3.3 - In Exercises 48-63, use the Gauss-Jordan method to...Ch. 3.3 - Prob. 60EQCh. 3.3 - Prob. 61EQCh. 3.3 - In Exercises 48-63, use the Gauss-Jordan method to...Ch. 3.3 - In Exercises 48-63, use the Gauss-Jordan method to...Ch. 3.4 - In Exercises 1 -6, solve the system Ax = b using...Ch. 3.4 - In Exercises 1 6, solve the system Ax = b using...Ch. 3.4 - In Exercises 1 -6, solve the system Ax = b using...Ch. 3.4 - In Exercises 1 -6, solve the system Ax = b using...Ch. 3.4 - In Exercises 1-6, solve the system Ax = b using...Ch. 3.4 - Prob. 6EQCh. 3.4 - In Exercises 7-12, find an LU factorization of the...Ch. 3.4 - In Exercises 7-12,find an LU factorization of the...Ch. 3.4 - In Exercises 7-12, find an LU factorization of the...Ch. 3.4 - In Exercises 7-12,find an LU factorization of the...Ch. 3.4 - In Exercises 7-12,find an LU factorization of the...Ch. 3.4 - Prob. 12EQCh. 3.4 - Generalize the definition of LU factorization to...Ch. 3.4 - Prob. 14EQCh. 3.5 - In Exercises 1-4, let S be the collection of...Ch. 3.5 - In Exercises 5-8, let S be the collection of...Ch. 3.5 - In Exercises 11 and 12, determine whether b is in...Ch. 3.5 - If A is the matrix in Exercise 12, is v=[712] in...Ch. 3.6 - 1. Let Ta : ℝ2 → ℝ2 be the matrix transformation...Ch. 3.6 - Let TA: 23 be the matrix transformation...Ch. 3.6 - In Exercises 3-6, prove that the given...Ch. 3.6 - In Exercises 3-6, prove that the given...Ch. 3.6 - Prob. 5EQCh. 3.6 - In Exercises 3-6, prove that the given...Ch. 3.6 - In Exercises 7-10, give a counterexample to show...Ch. 3.6 - In Exercises 7-10, give a counterexample to show...Ch. 3.6 - In Exercises 7-10, give a counterexample to show...Ch. 3.6 - In Exercises 7-10, give a counterexample to show...Ch. 3.6 - In Exercises 11-14, find the standard matrix of...Ch. 3.6 - In Exercises 11-14, find the standard matrix of...Ch. 3.6 - In Exercises 11-14, find the standard matrix of...Ch. 3.6 - In Exercises 11-14, find the standard matrix of...Ch. 3.6 - In Exercises 15-18, show that the given...Ch. 3.6 - In Exercises 15-18, show that the given...Ch. 3.6 - Prob. 17EQCh. 3.6 - Prob. 18EQCh. 3.6 - In Exercises 20-25, find the standard matrix of...Ch. 3.6 - In Exercises 20-25, find the standard matrix of...Ch. 3.6 - In Exercises 20-25, find the standard matrix of...Ch. 3.6 - In Exercises 20-25, find the standard matrix of...Ch. 3.6 - In Exercises 20-25, find the standard matrix of...Ch. 3.6 - In Exercises 20-25, find the standard matrix of...Ch. 3.6 - In Exercises30-35, verify Theorem 3.32 by finding...Ch. 3.6 - In Exercises 30-35, verify Theorem 3.32 by finding...Ch. 3.6 - In Exercises 30-35, verify Theorem 3.32 by finding...Ch. 3.6 - In Exercises 30-35, verify Theorem 3.32 by finding...Ch. 3.6 - In Exercises30-35, verify Theorem 3.32 by finding...Ch. 3.6 - Prob. 35EQCh. 3.7 - In Exercises 1-4, let be the transition matrix...Ch. 3.7 - Prob. 2EQCh. 3.7 - In Exercises 1-4, let P=[0.50.30.50.7] be the...Ch. 3.7 - In Exercises 1-4, let be the transition matrix for...Ch. 3.7 - Prob. 5EQCh. 3.7 - Prob. 6EQCh. 3.7 - Prob. 7EQCh. 3.7 - Prob. 8EQCh. 3.7 -
12. Robots have been programmed to traverse the...Ch. 3.7 - Prob. 31EQCh. 3.7 - Prob. 32EQCh. 3.7 - Prob. 33EQCh. 3.7 - Prob. 34EQCh. 3.7 - Prob. 35EQCh. 3.7 - Prob. 36EQCh. 3.7 - Prob. 37EQCh. 3.7 - Prob. 38EQCh. 3.7 - Prob. 39EQCh. 3.7 - Prob. 40EQCh. 3.7 - In Exercises 45-48, determine the adjacency matrix...Ch. 3.7 - Prob. 46EQCh. 3.7 - In Exercises 45-48, determine the adjacency matrix...Ch. 3.7 - In Exercises 45-48, determine the adjacency matrix...Ch. 3.7 - Prob. 53EQCh. 3.7 - In Exercises 53-56, determine the adjacency matrix...Ch. 3.7 - In Exercises 53-56, determine the adjacency matrix...Ch. 3.7 - In Exercises 53-56, determine the adjacency matrix...
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