Linear Algebra: A Modern Introduction
4th Edition
ISBN: 9781285463247
Author: David Poole
Publisher: Cengage Learning
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Chapter 3.2, Problem 12EQ
In Exercises 9-12, find the general form of the span of the indicated matrices, as in Example 3.17.
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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
In Exercises 1-16, compute the indicated...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
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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- Show that the matrix below does not have an LU factorization. A=0110arrow_forwardIn Exercises 9-12, find the general form of the span of the indicated matrices, as in Example 3.17. span (A1,A2) in Exercise 5arrow_forwardCan a matrix with zeros on the diagonal have an inverse? If so, find an example. If not, prove why not. For simplicity, assume a 22 matrix.arrow_forward
- Illustrate the result of Exercise 63 with the matrix A=[211312022]arrow_forwardIn general, it is difficult to show that two matrices are similar. However, if two similar matrices are diagonalizable, the task becomes easier. In Exercises 38-41, show that A and B are similar by showing that they are similar to the same diagonal matrix. Then find an invertible matrix P such that .arrow_forwardCan we add any two matrices together? If so, explain why; if not, explain why not and give an example of two matrices that cannot be added together.arrow_forward
- Use an example chosen from 22 matrices to show that for nn matrices A and B,ABBA but AB=BA.arrow_forwardDetermine if the statement is true or false. If the statement is false, then correct it and make it true. For the product of two matrices to be defined, the number of rows of the first matrix must equal the number of columns of the second matrix.arrow_forwardConsider the matrix A=[2314]. Show that any of the three types of elementary row operations can be used to create a leading 1 at the top of the first column. Which do you prefer and why?arrow_forward
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