A modern introduction to linear algebra

Author

Introduction

1. Vectors

1.1. Vectors in R n

1.1.1. Euclidean n-Space

1.1.2. Vector Addition/Subtraction

1.1.3. Scalar Multiplication

1.1.4. Geometric Vectors in R 2 and R 3

1.1.5. Algebraic Properties

1.2. The Inner Product and Norm

1.2.1. The Angle between Vectors

1.3. Spanning Sets

1.4. Linear Independence

1.5. Bases

1.5.1. Coordinates/Change of Basis

1.6. Subspaces

1.7. Summary

2. Systems of Equations

2.1. The Geometry of Systems of Equations in R 2 and R 3

2.2. Matrices and Echelon Form

2.2.1. The Matrix of Coefficients

2.2.2. Elementary Row Operations

2.3. Gaussian Elimination

*2.4. Computational Considerations-Pivoting

2.5. Gauss-Jordan Elimination and Reduced Row Echelon Form

*2.6. Ill-Conditioned Systems of Linear Equations

2.7. Rank and Nullity of a Matrix

2.7.1. Row Spaces and Column Spaces

2.7.2. The Null Space

2.8. Systems of m Linear Equations in n Unknowns

2.8.1. The Solution of Linear Systems

2.9. Summary

3. Matrix Algebra

3.1. Addition and Subtraction of Matrices

3.1.1. Scalar Multiplication

3.1.2. Transpose of a Matrix

3.2. Matrix-Vector Multiplication

3.2.1. Matrix-Vector Multiplication as a Transformation

3.3. The Product of Two Matrices

3.3.1. The Column View of Multiplication

3.3.2. The Row View of Multiplication

3.3.3. Positive Powers of a Square Matrix

3.4. Partitioned Matrices

3.5. Inverses of Matrices

3.5.1. Negative Powers of a Square Matrix

3.6. Elementary Matrices

3.7. The LU Factorization

3.8. Summary

4. Eigenvalues, Eigenvectors, and Diagonalization

4.1. Determinants

4.1.1. Cramer's Rule

*4.2. Determinants and Geometry

4.3. The Manual Calculation of Determinants

4.4. Eigenvalues and Eigenvectors

4.5. Similar Matrices and Diagonalization

4.6. Algebraic and Geometric Multiplicities of Eigenvalues

*4.7. The Diagonalization of Real Symmetric Matrices

4.8. The Cayley-Hamilton Theorem (a First Look)/the Minimal Polynomial

4.9. Summary

5. Vector Spaces

5.1. Vector Spaces

5.2. Subspaces

5.2.1. The Sum and Intersection of Subspaces

5.3. Linear Independence and the Span

5.4. Bases and Dimension

5.5. Summary

6. Linear Transformations

6.1. Linear Transformations

6.2. The Range and Null Space of a Linear Transformation

6.3. The Algebra of Linear Transformations

6.4. Matrix Representation of a Linear Transformation

6.5. Invertible Linear Transformations

6.6. Isomorphisms

6.7. Similarity

6.8. Similarity Invariants of Operators

6.9. Summary

7. Inner Product Spaces

7.1. Complex Vector Spaces

7.2. Inner Products

7.3. Orthogonality and Orthonormal Bases

7.4. The Gram-Schmidt Process

7.5. Unitary Matrices and Orthogonal Matrices

7.6. Schur Factorization and the Cayley-Hamilton Theorem

7.7. The QR Factorization and Applications

7.8. Orthogonal Complements

7.9. Projections

7.10. Summary

8. Hermitian Matrices and Quadratic Forms

8.1. Linear Functionals and the Adjoint of an Operator

8.2. Hermitian Matrices

8.3. Normal Matrices

*8.4. Quadratic Forms

*8.5. Singular Value Decomposition

8.6. The Polar Decomposition

8.7. Summary

Appendix A. Basics of Set Theory

Appendix B. Summation and Product Notation

Appendix C. Mathematical Induction

Appendix D. Complex Numbers

Answers/Hints to Odd-Numbered Problems

Index