**Basic linear algebra:**

Vector space and subspace

Basis and dimension

Null space and range

Rank

Norm and inner product

**Orthogonality:**

Orthogonality and orthogonal projection

Gram–Schmidt procedure and QR decomposition

**Systems of linear equations:**

Inverse

Least square solution of overdetermined equations

Least norm solution of underdetermined equations

**Eigenvalue decomposition:**

Eigenvalues and eigenvectors

Jordan canonical form

**Hermitian and symmetric matrices:**

Quadratic forms

Positive semidefinite matrices

**Singular value decomposition:**

Singular values and singular vectors

Matrix norms

Low-rank matrix approximation

**Advanced topics:**

Perron–Frobenius theory

Toeplitz and circulant matrices

Linear dynamical systems

Linear and nonlinear MMSE estimation

Lattices

Sparse signal processing

Clustering

MIMO communication

Coding theory

Markov chains

PageRank algorithm

Spectral graph theory

Combinatorics

This is a tentative schedule for the course. Changes may be made as the
course progresses, and we will update the lecture topics accordingly.

[Jan 07, Mon] Lecture #1: Introduction

[Jan 09, Wed] Lecture #2: Vector spaces and subspaces

[Jan 11, Fri] Lecture #3: Basis and dimension

[Jan 14, Mon] Lecture #4: Linear transformations and matrices

[Jan 16, Wed] Lecture #5: Rank and nullity

[Jan 18, Fri] Lecture #6: Inverses and change of basis

[Jan 21, Mon] No lecture (Martin Luther King Junior Day)

[Jan 23, Wed] Lecture #7: Inner product and orthogonal projection

[Jan 25, Fri] Lecture #8: Orthogonal matrices

[Jan 28, Mon] Lecture #9: QR decomposition and Gram-Schmidt procedure

[Jan 30, Wed] Lecture #10: Orthogonal decomposition of a vector space

[Feb 01, Fri] Midterm #1

[Feb 04, Mon] Lecture #11: Systems of linear equations and inverses

[Feb 06, Wed] Lecture #12: Least-squares solutions

[Feb 08, Fri] Lecture #13: Least-norm solutions and pseudoinverses

[Feb 11, Mon] Lecture #14: Sparsest solutions

[Feb 13, Wed] Lecture #15: Eigenvalues and eigenvectors

[Feb 15, Fri] Lecture #16: Jordan canonical forms

[Feb 18, Mon] No lecture (Presidents’ Day)

[Feb 20, Wed] Lecture #17: Linear dynamical systems

[Feb 22, Fri] Lecture #18: Cayley-Hamilton theorem and matrix functions

[Feb 25, Mon] Midterm #2

[Feb 27, Wed] Lecture #19: Nonnegative matrices and Perron-Frobenius theory

[Mar 01, Fri] Lecture #20: Markov chains and random walk on graphs

[Mar 04, Mon] Lecture #21: Hermitian and symmetric matrices

[Mar 06, Wed] Lecture #22: Positive semidefinite matrices and Cholesky decomposition

[Mar 08, Fri] Lecture #23: Sylvestor's criterion and ellipsoids

[Mar 11, Mon] Lecture #24: Singular values and singular vectors

[Mar 13, Wed] Lecture #25: Matrix norm and pseudoinverse revisited

[Mar 15, Fri] Lecture #26: Low-rank approximation and principal component analysis

[Mar 20, Wed] Final exam