The core linear algebra material consists of 6 units:
- Vectors, Matrices, Inverses, and Determinants: Vectors, operations with vectors, dot product, norms, matrices, operations with matrices, special matrices, transpose, inverses, determinants, gradient vectors.
- Linear Equations, and Elimination: Linear equations, Gaussian elimination, pivots, rank, LU factorization, Taylor series expansion with Jacobians.
- Linear Transformations and Vector Spaces: Linear transformations, the matrix of a linear transformation, vector spaces, abstract vector spaces, range, null space, and the other two spaces of a matrix, relationships between the four fundamental subspaces, linearization, change of variables in multivariate integration.
- Orthogonality, Projections, and Least Squares: Orthogonal and orthonormal vectors, orthogonal vector spaces, orthogonal relationships for the fundamental subspaces, Gram-Schmidt, QR factorization, projections, least squares in various forms.
- Spectral Properties and Quadratic Forms: Eigenvalues and eigenvectors, diagonalization and the similarity transform, matrix powers and modes, matrix exponentials, spectral properties of symmetric matrices quadratic forms, positive definite (semi-definite) matrices, hessians and convexity.
- Singular Value Decomposition: SVD, Matrix norms, Eckart-Young, conditions number, least squares via SVD.
Jupyter notebooks for each of these units is updated to this
GitHub repo during the course.