The readings are assigned in: Strang, Gilbert. *Introduction to Linear Algebra*. 4th ed. Wellesley, MA: Wellesley-Cambridge Press, February 2009. ISBN: 9780980232714. The Table of Contents is listed below.

Course readings.SES # | TOPICS | READINGS |
---|

1 | The geometry of linear equations | 1.1-2.1 |

2 | Elimination with matrices | 2.2-2.3 |

3 | Matrix operations and inverses | 2.4-2.5 |

4 | *LU* and *LDU *factorization | 2.6 |

5 | Transposes and permutations | 2.7 |

6 | Vector spaces and subspaces | 3.1 |

7 | The nullspace: Solving Ax = 0 | 3.2 |

8 | Rectangular *PA *= *LU* and Ax = b | 3.3-3.4 |

9 | Row reduced echelon form | 3.3-3.4 |

10 | Basis and dimension | 3.5 |

11 | The four fundamental subspaces | 3.6 |

12 | Exam 1: Chapters 1 to 3.4 | |

13 | Graphs and networks | 8.2 |

14 | Orthogonality | 4.1 |

15 | Projections and subspaces | 4.2 |

16 | Least squares approximations | 4.3 |

17 | Gram-Schmidt and *A* = *QR* | 4.4 |

18 | Properties of determinants | 5.1 |

19 | Formulas for determinants | 5.2 |

20 | Applications of determinants | 5.3 |

21 | Eigenvalues and eigenvectors | 6.1 |

22 | Diagonalization | 6.2 |

23 | Markov matrices | 8.3 |

24 | *Review for exam 2* | |

25 | Exam 2: Chapters 1-5, 6.1-6.2, 8.2 | |

26 | Differential equations | 6.3 |

27 | Symmetric matrices | 6.4 |

28 | Positive definite matrices | 6.5 |

29 | Matrices in engineering | 8.1 |

30 | Similar matrices | 6.6 |

31 | Singular value decomposition | 6.7 |

32 | Fourier series, FFT, complex matrices | 8.5, 10.2-10.3 |

33 | Linear transformations | 7.1-7.2 |

34 | Choice of basis | 7.3 |

35 | Linear programming | 8.4 |

36 | *Course review* | |

37 | Exam 3: Chapters 1-8 (8.1, 2, 3, 5) | |

38 | Numerical linear algebra | 9.1-9.3 |

39 | Computational science | See the Web site for 18.085 |

40 | Final exam | |

## Table of Contents

### 1. Introduction to Vectors

1.1 Vectors and Linear Combinations

1.2 Lengths and Dot Products

1.3 Matrices

### 2. Solving Linear Equations

2.1 Vectors and Linear Equations

2.2 The Idea of Elimination

2.3 Elimination Using Matrices

2.4 Rules for Matrix Operations

2.5 Inverse Matrices

2.6 Elimination = Factorization: A = LU

2.7 Transposes and Permutations

### 3. Vector Spaces and Subspaces

3.1 Spaces of Vectors

3.2 The Nullspace of *A*: Solving Ax = 0

3.3 The Rank and the Row Reduced Form

3.4 The Complete Solution to Ax = b

3.5 Independence, Basis, and Dimension

3.6 Dimensions of the Four Subspaces

### 4. Orthogonality

4.1 Orthogonality of the Four Subspaces

4.2 Projections

4.3 Least Squares Approximations

4.4 Orthogonal Bases and Gram-Schmidt

### 5. Determinants

5.1 The Properties of Determinants

5.2 Permutations and Cofactors

5.3 Cramer's Rule, Inverses, and Volumes

### 6. Eigenvalues and Eigenvectors

6.1 Introduction to Eigenvalues

6.2 Diagonalizing a Matrix

6.3 Applications to Differential Equations

6.4 Symmetric Matrices

6.5 Positive Definite Matrices

6.6 Similar Matrices

6.7 Singular Value Decomposition (SVD)

### 7. Linear Transformations

7.1 The Idea of a Linear Transformation

7.2 The Matrix of a Linear Transformation

7.3 Diagonalization and the Pseudoinverse

### 8. Applications

8.1 Matrices in Engineering

8.2 Graphs and Networks

8.3 Markov Matrices, Population, and Economics

8.4 Linear Programming

8.5 Fourier Series: Linear Algebra for Functions

8.6 Linear Algebra for Statistics and Probability

8.7 Computer Graphics

### 9. Numerical Linear Algebra

9.1 Gaussian Elimination in Practice

9.2 Norms and Condition Numbers

9.3 Iterative Methods and Preconditioners

### 10. Complex Vectors and Complex Matrices

10.1 Complex Numbers

10.2 Hermitian and Unitary Matrices

10.3 The Fast Fourier Transform