- Linear Algebra with Applications
- Dedication
- Contents
- Preface
- Chapter 1 Matrices and Systems of Equations
- Chapter 2 Determinants
- Chapter 3 Vector Spaces
- Chapter 4 Linear Transformations
- Chapter 5 Orthogonality
- 5.1 The Scalar Product in ℝn
- Section 5.1Exercises
- 5.2 Orthogonal Subspaces
- Section 5.2 Exercises
- 5.3 Least Squares Problems
- Section 5.3 Exercises
- 5.4 Inner Product Spaces
- Section 5.4 Exercises
- 5.5 Orthonormal Sets
- Section 5.5 Exercises
- 5.6 The Gram–Schmidt Orthogonalization ProcessThe Gram–Schmidt Orthogonalization Process
- Section 5.6 Exercises
- 5.7 Orthogonal Polynomials
- Section 5.7 Exercises
- Chapter 5 Exercises
- Chapter 6 Eigenvalues
- 6.1 Eigenvalues and Eigenvectors
- Section 6.1 Exercises
- 6.2 Systems of Linear Differential Equations
- Section 6.2 Exercises
- 6.3 Diagonalization
- Section 6.3 Exercises
- 6.4 Hermitian Matrices
- Section 6.4 Exercises
- 6.5 The Singular Value Decomposition
- Section 6.5 Exercises
- 6.6 Quadratic Forms
- Section 6.6 Exercises
- 6.7 Positive Definite Matrices
- Section 6.7 Exercises
- 6.8 Nonnegative Matrices
- Section 6.8 Exercises
- Chapter 6 Exercises
- Chapter 7 Numerical Linear Algebra
- 7.1 Floating-Point Numbers
- Section 7.1 Exercises
- 7.2 Gaussian Elimination
- Section 7.2 Exercises
- 7.3 Pivoting Strategies
- Section 7.3 Exercises
- 7.4 Matrix Norms and Condition Numbers
- Section 7.4 Exercises
- 7.5 Orthogonal Transformations
- Section 7.5 Exercises
- 7.6 The Eigenvalue Problem
- Section 7.6 Exercises
- 7.7 Least Squares Problems
- Section 7.7 Exercises
- 7.8 Iterative Methods
- Section 7.8 Exercises
- Chapter 7 Exercises
- Chapter 8 Canonical Forms
- Appendix: Matlab
- Bibliography
- Answers to Selected Exercises
- Index
Chapter Test B
1. Hint: The computation of (AB)x involves a matrix-matrix multiplication followed by a matrix-vector multiplication.The multiplication A(Bx) should be more efficient since it only involves 2 matrix-vector multiplications.
3. Hint: Show that the singular values of Q are all equal to 1.
5. Hint: If 15 digit decimal arithmetic is used the machine epsilon will be
0.5×10−4 .7. Hint:
B=QTAQ
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