Let A = 0 3 1 1 2 2 2 5 4 and b = 1 7 1 (a) Reorder the rows of (A|b) in the order (2, 3, 1) and then solve the reordered system. (b) Factor A into a product PTLU, where P is the permutation matrix corresponding to the reordering in part (a).
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Table of Contents
1
Matrices and Systems of Equations
1.1
Systems of Linear Equations
1.2
Row Echelon Form
1.3
Matrix Arithmetic
1.4
Matrix Algebra
1.5
Elementary Matrices
1.6
Partitioned Matrices
2
Determinants
2.1
The Determinant of a Matrix
2.2
Properties of Determinants
2.3
Additional Topics and Applications
3
Vector Spaces
3.1
Definition and Examples
3.2
Subspaces
3.3
Linear Independence
3.4
Basis and Dimension
3.5
Change of Basis
3.6
Row Space and Column Space
4
Linear Transformations
4.1
Definition and Examples
4.2
Matrix Representations of Linear Transformations
4.3
Similarity
5
Orthogonality
5.1
The Scalar Product in Rn
5.2
Orthogonal Subspaces
5.3
Least Squares Problems
5.4
Inner Product Spaces
5.5
Orthonormal Sets
5.6
The GramSchmidt Orthogonalization Process
5.7
Orthogonal Polynomials
6
Eigenvalues
6.1
Eigenvalues and Eigenvectors
6.2
Systems of Linear Differential Equations
6.3
Diagonalization
6.4
Hermitian Matrices
6.5
The Singular Value Decomposition
6.6
Quadratic Forms
6.7
Positive Definite Matrices
6.8
Nonnegative Matrices
7
Numerical Linear Algebra
7.1
Floating-Point Numbers
7.2
Gaussian Elimination
7.3
Pivoting Strategies
7.4
Matrix Norms and Condition Numbers
7.5
Orthogonal Transformations
7.6
The Eigenvalue Problem
7.7
Least Squares Problems
Textbook Solutions for Linear Algebra with Applications
Chapter 7.3 Problem 4
Question
Let A = 3 2 2 4 and b = 5 2 Solve the system Ax = b using complete pivoting. Let P be the permutation matrix determined by the pivot rows and Q the permutation matrix de termined by the pivot columns. Factor PAQ into a product LU.
Solution
The first step in solving 7.3 problem number 4 trying to solve the problem we have to refer to the textbook question: Let A = 3 2 2 4 and b = 5 2 Solve the system Ax = b using complete pivoting. Let P be the permutation matrix determined by the pivot rows and Q the permutation matrix de termined by the pivot columns. Factor PAQ into a product LU.
From the textbook chapter Pivoting Strategies you will find a few key concepts needed to solve this.
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full solution
full solution
Title
Linear Algebra with Applications 8
Author
Steve Leon
ISBN
9780136009290
Let A = 3 2 2 4 and b = 5 2 Solve the system Ax = b using complete pivoting. Let P be
Chapter 7.3 textbook questions
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Chapter 7: Problem 1 Linear Algebra with Applications 8
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Chapter 7: Problem 2 Linear Algebra with Applications 8
Let A be the matrix in Exercise 1. Use the factorization PTLU to solve Ax = c for each of the following choices of c: (a) (8, 1, 20)T (b) (9,2,7)T (c) (4, 1, 11)T
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Chapter 7: Problem 3 Linear Algebra with Applications 8
Let A = 1 8 6 1 4 5 2 4 6 and b = 8 1 4 Solve the system Ax = b using partial pivoting. If P is the permutation matrix corresponding to the pivoting strategy, factor PA into a product LU.
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Chapter 7: Problem 4 Linear Algebra with Applications 8
Let A = 3 2 2 4 and b = 5 2 Solve the system Ax = b using complete pivoting. Let P be the permutation matrix determined by the pivot rows and Q the permutation matrix de termined by the pivot columns. Factor PAQ into a product LU.
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Chapter 7: Problem 5 Linear Algebra with Applications 8
Let A be the matrix in Exercise 4 and let c = (6,4)T . Solve the system Ax = c in two steps: (a) Set z = QT x and solve LUz = Pc for z. (b) Calculate x = Qz.
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Chapter 7: Problem 6 Linear Algebra with Applications 8
Let A = 5 4 7 2 4 3 2 8 6 , b = 2 5 4 , c = 5 4 2 (a) Use complete pivoting to solve the system Ax = b. (b) Let P be the permutation matrix determined by the pivot rows, and let Q be the permutation matrix determined by the pivot columns. Factor PAQ into a product LU. (c) Use the LU factorization from part (b) to solve the system Ax = c.
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Chapter 7: Problem 7 Linear Algebra with Applications 8
The exact solution of the system 0.6000x1 + 2000x2 = 2003 0.3076x1 0.4010x2 = 1.137 is x = (5, 1)T . Suppose that the calculated value of x2 is x_ 2 = 1+_. Use this value in the first equation and solve for x1. What will the error be? Calculate the relative error in x1 if _ = 0.001.
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Chapter 7: Problem 8 Linear Algebra with Applications 8
Solve the system in Exercise 7 using four-digit decimal floating-point arithmetic and Gaussian elimination with partial pivoting.
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Chapter 7: Problem 9 Linear Algebra with Applications 8
Solve the system in Exercise 7 using four-digit decimal floating-point arithmetic and Gaussian elimination with complete pivoting. 1
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Chapter 7: Problem 10 Linear Algebra with Applications 8
Use four-digit decimal floating-point arithmetic, and scale the system in Exercise 7 by multiplying the first equation through by 1/2000 and the second equation through by 1/0.4010. Solve the scaled system using partial pivoting.
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