# ECE269: Linear Algebra and Applications

## Course topics (mathematical foundation)

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:

• Positive semidefinite matrices

Singular value decomposition:

• Singular values and singular vectors

• Matrix norms

• Low-rank matrix approximation

• Perron–Frobenius theory

• Toeplitz and circulant matrices

## Course topics (applications)

• 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

## Course schedule

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-norm solutions and pseudoinverses

• [Feb 08, Fri] Lecture #13: Least-squares solutions

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

• [Feb 13, Wed] Lecture #15: Eigenvalue decomposition and Jorcan canonical forms

• [Feb 15, Fri] Lecture #16: Linear dynamical systems

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

• [Feb 20, Wed] Lecture #17: Cayley-Hamilton theorem and matrix functions

• [Feb 22, Fri] Midterm #2

• [Feb 25, Mon] Lecture #18: Nonnegative matrices and Perron-Frobenius theory

• [Feb 27, Wed] Lecture #19: Markov chains and random walk on graphs

• [Mar 01, Fri] Lecture #20: Hermitian and symmetric matrices

• [Mar 04, Mon] Lecture #21: Rayleigh quotients and definite matrices

• [Mar 06, Wed] Lecture #22: Cholesky decomposition, Sylvestor's criterion and ellipsoids

• [Mar 08, Fri] Lecture #23: Singular values and singular vectors

• [Mar 11, Mon] Lecture #24: Matrix norm and pseudoinverse revisited

• [Mar 13, Wed] Lecture #25: Low-rank approximation and principal component analysis

• [Mar 15, Fri] Lecture #26: Final review session

• [Mar 20, Wed] Final exam