- 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 A
1. Hint: See Theorem2.2.3 .
2. Hint: Try some examples with
2×2 matrices.3. Hint: See Exercise5 in Section 2.2.
4. Hint: See Theorems2.1.2 and 2.2.3.
5. Hint: An easy way to make up a counterexample would be to use diagonal matrices for A and B.
6. Hint: Make use of Theorem2.2.3 to prove the result.
7. Hint: See Theorems2.1.3 and 2.2.2.
8. Hint: See Theorem 1.5.2.
9. Hint: What are the effects of row operations I and II on the value of the determinant?
10. Hint: How is
det(Ak) related todet(A)?
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