Linear Algebra

I have taught and coordinated Linear Algebra over the course of four years. The lecture notes have changed drastically over these years, and below you will find the most recent version of the notes. You will also find a few supplementary videos that go with some of the lectures. Any lecture that is marked with an asterisk has supplementary videos.

Course Calendar

Lecture Notes

Lecture 1: Vectors, Linear Combinations of Vectors, Dot Products (magnitude and angles between vectors)

Lecture 2: Matrices and Column Spaces

Lecture 3*: Matrix Multiplication, A=CR Factorization

Lecture 4*: Elimination, Elimination Matrices, Permutation Matrices

Lecture 5: Inverse of a Matrix, Transpose of a Matrix

Lecture 6*: A=LU and PA=LU Factorizations, Symmetric Matrices

Lecture 7: Vector Spaces and Subspaces

Lecture 8: The Nullspace of a matrix, Row-Echelon Form (REF), Reduced Row-Echelon Form (RREF)

Lecture 9*: The Complete Solution to a System Ax=b, Rank and Solvability

Lecture 10: Basis and Dimension of a Vector Space

Lecture 11*: Dimensions of the Four Fundamental Subspaces, Orthogonality of the Four Subspaces

Lecture 12: Projections onto Lines and Subspaces

Lecture 13: Least Squares Approximation

Lecture 14: Orthogonal Matrices, Orthogonal Bases, Gram-Schmidt, A=QR Factorization

Lecture 15*: 3 by 3 Determinants, Properties and Applications of Determinants

Lecture 16*: Linear Transformations

Lecture 17: Linear Transformations (Change of Basis)

Lecture 18: Introduction to Eigenvalues

Lecture 19: Introduction to Linear Algebra in Python

Lecture 20: Diagonalizing a Matrix

Lecture 21*: Symmetric Positive Definite Matrices

Lecture 22: Singular Value Decomposition (SVD): Compressing Images by the SVD

Lecture 23: Singular Value Decomposition (SVD): Connections to the Four Fundamental Subspaces

Supplementary Lecture Videos (only for the lectures with *)

Matrix Operations

Elimination and Permutation Matrices

LU Factorization

LDU Factorization

Symmetric Matrices

PA=LU Decomposition

Complete Solutions

Full Rank

The Four Fundamental Subspaces

Orthogonality of the Four Subspaces and the Fundamental Theorem of Linear Algebra


Properties of Determinants

Formulas to Compute Determinants

Cofactor Method to Compute Determinants

Linear Transformations

Matrix of a Linear Transformation