Lecture Notes
Antoine Cerfon
CIMS, NYU
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Notes for undergraduate Linear Algebra
The notes below follow closely the textbook
Introduction to Linear Algebra, Fourth Edition
by Gilbert Strang.
Lecture 1: Vectors and Linear Combinations
Lecture 2: Lengths and the dot product
Lecture 3: A Brief Introduction to Matrices
Lecture 4: Solving Linear Systems
Lecture 5: Elimination in terms of Matrix Operations
Lecture 6: Inverse Matrices
Lecture 7: LU Factorization
Lecture 8: Transposes and Permutations
Lecture 9: Vector Spaces
Lecture 10: The Nullspace of a Matrix and Solving Ax=0
Lecture 11: Solving Ax=b in general
Lecture 12: Independence, Basis and Dimension
Lecture 13: Dimensions of the Four Subspaces
Lecture 14: Orthogonality of the Four Subspaces
Lecture 15: Projections
Lecture 16: Least Squares Approximations
Lecture 17: Orthogonal Bases, Orthogonal Matrices, and QR Decomposition
Lecture 18: Introduction to Determinants
Lecture 19: Permutations and Cofactors
Lecture 20: Cramer's rule, Inverses, and Volumes
Lecture 21: Introduction to Eigenvectors and Eigenvalues
Lecture 22: Diagonalizing a Matrix
Lecture 23: Symmetric Matrices
Lecture 24: Positive Definite Matrices
Lecture 25: Similar Matrices
Lecture 26: Singular Value Decomposition (SVD)
Lecture 27: Linear Transformations, and their Matrix Representation