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. |