This is a set of lecture notes and problems I created for Introduction to Linear Algebra taught in Spring 2003 at Applied Mathematics and Statistics Department at Stony Brook University. There are many typos, as I have never fully proof-read them. Please let me know if you find them useful

*Lecture Notes*

Lecture 01 | Introduction. Numbers. Sets. |

Lecture 02 | Linear Equations. |

Lecture 03 | Linear Systems. Row Echelon Form. Gaussian Elimination. |

Lecture 04 | Overview of Linear Systems. |

Lecture 05 | Matrices and Matrix Operations. |

Lecture 06 | Inverse and Transpose. Matrices and Linear Systems. |

Lecture 07 | Matrix Equations and the Inverse. |

Lecture 08 | Vector Spaces and Subspaces. |

Lecture 09 | Linear Combinations. Linear Dependence. |

Lecture 10 | Spanning Sets. Basis. |

Lecture 11 | Examples of Bases. Dimension. |

Lecture 12 | More Examples. Dimension and Basis of the Span. |

Lecture 13 | Dimensions and Basis of the Span. Rank. |

Lecture 14 | Functions. Linear Functions. |

Lecture 15 | Homogeneous Systems. |

Lecture 16 | Image and Kernel. Matrix of a Linear Function. |

Lecture 17 | Dimension and Basis of Image and Kernel. |

Lecture 18 | Image and Kernel and Matrices. Linear Functions as a Space. |

Lecture 19 | Area of a Parallelogram. |

Lecture 20 | Permutations. |

Lecture 21 | General Properties of Area and Volume. Determinant. |

Lecture 22 | Properties of Determinants – 1. |

Lecture 23 | Properties of Determinants – 2. |

Lecture 23 – Addendum | Proofs. |

Lecture 24 | Application of Determinants. Kramer’s Rule. Inverse. |

Lecture 25 | Euclidean Spaces. Norm. Cauchy Inequality. |

Lecture 26 | Orthogonality. |

Lecture 27 | Orthogonal Bases. Gram-Schmidt Process. |

Lecture 28 | Operators. Change of Basis. Matrix of an Operator. |

Lecture 29 | Change of Matrix of an Operator. Diagonalizable Operators. |

Lecture 30 | Eigenvalues and Eigenvectors. |

Lecture 31 | Symmetric Matrices. |

Lecture 32 | Powers and Square Roots of Matrices. |

Lecture 33 | Invariant Spaces. Jordan Canonical Form. |

Lecture 34 | Functions of Operators |

*Problem Sets*

Problem Set 1 | Linear Systems. |

Problem Set 2 | Matrices. |

Problem Set 3 | Vector Spaces. Bases and Dimensions. |

Problem Set 4 | Linear Functions. |

Problem Set 5 | Determinants. |

Problem Set 6 | Euclidean Spaces. Orthogonality. Norms. |

Problem Set 7 | Linear Operators. Eigenvalues and Eigenvectors. |