Course Syllabus

Table of contents

1. Applications in Data Science
2. System of Linear Equations and Matrices
3. Matrices
4. Vectors
5. Vector Spaces
6. Matrices and Spaces
7. Linear Transformations
8. Determinants
9. Orthogonality
10. Eigen-Theory
11. Quadratic form
12. Singular Value Decomposition

Applications in Data Science

Least square problems; word embedding; graph adjacency matrix; portfolio management; convolution; page rank and web search; recommendation system and matrix completion; topic model.

System of Linear Equations and Matrices

Gaussian elimination, row operations, (reduced) row-echelon form, equivalent systems, solution set, augmented matrices, geometric interpretation.

Matrices

Matrix operations, special matrices, partitioned matrices, matrix transpose, symmetric matrices, matrix inverse, nonsingular matrices, row operations and elementary matrices, LU decomposition.

Vectors

Vector operations, linear combinations, linear span, linear independence, spanning sets.

Vector Spaces

Vector spaces, subspaces, basis and dimension, change of basis.

Matrices and Spaces

Column and row spaces, null spaces, rank of matrices, matrix space.

Linear Transformations

Kernel, image, range, basis change, matrix representations of linear transformations, similarity.

Determinants

Determinants, minors and cofactors, adjoint matrix and matrix inverse, Cramer’s rule.

Orthogonality

Orthogonal subspaces, projections, orthonormal basis, Gram-Schmidt process, QR decomposition; function space, Fourier series.

Eigen-Theory

Eigenvalues, eigenvectors, trace, page rank and web search; characteristic polynomial, diagonalization, eigenvalue decomposition, spectral theorem. Non-dignolizable matrices, complex eigenvalues, Hermitian matrix.

Quadratic form

Quadratic forms, positive definite matrices.

Singular Value Decomposition

Singular value decomposition, four fundamental subspaces; low rank approximation.