Math for Intelligent Systems, COT5615

COT 5615, Math for Intelligent Systems, Fall 2025

Place:MCCA; G186
Time:Monday, Wednesday, Friday 8 (3:00-3:50 p.m.)

Instructor:
Arunava Banerjee
Office: MALA 6101.
E-mail: arunava@ufl.edu.
Office hours (On Zoom-- 924 861 2325): TBD.

TAs:
Ke Chen
TA Office hours: (On Zoom-- ): TBD.
Tinghui Zhang
TA Office hours: (On Zoom-- ): TBD.

Catalog Description:

COT 5615: Mathematics for Intelligent Systems

Credits: 3 Grading Scheme: Letter

Prerequisite: MAC 2313, Multivariate Calculus; MAS 3114 or MAS 4105, Linear Algebra; STA 4321, Mathematical Statistics.

Mathematical methods commonly used to develop algorithms for computer systems that exhibit intelligent behavior.

Course Objectives:

The goal of this course is to cover several topics in mathematics that is of general interest to people pursuing a Ph.d in intelligent systems. The course will focus on conceptual clarity.

This course is an official pre-requisite for CAP6610 (Machine learning)

Required Text:

There is no official text book for this course. We will mostly work with material posted online. However, following are four good books to have.

References:
Principles of Mathematical Analysis, W. Rudin
Linear Algebra, K. M. Hoffman, R. Kunze, can be found online at https://www.math.pku.edu.cn/teachers/anjp/textbook.pdf
Probability and Measure, P. Billingsley
Probability: Theory and examples, R. Durrett, can be found online at https://services.math.duke.edu/~rtd/PTE/PTE5_011119.pdf

Please return to this page at least once a week to check updates in the table below

Evaluation: The final grade will be based on three midterm exams (25% each) and several assignments (remaining 25%).

Course Policies:

Late assignments: All homework assignments are due before class.
Plagiarism: You are expected to submit your own solutions to the assignments. Feel free to discuss the concepts underlying the questions with friends and classmates.
Attendance: Their is no official attendance requirement. If you find better use of the time spent sitting thru lectures, please feel free to devote such to any occupation of your liking. However, keep in mind that it is your responsibility to stay abreast of the material presented in class.
Cell Phones: Please, no phone calls during class. Please turn off the ringer on your cell phone before coming to class.
University Honesty Policy, Campus Resources and other important information : See here for grad and here for undergrad.

Tentative List of Topics to be covered

Real analysis: Rationals and Reals, Basic point set topology, convergence, continuity etc (approximately Chap 1-4 of Rudin)
Vector spaces and Linear algebra (approximately Chap 1-3, 8 of Hoff. Kun.)
Mathematical Probability theory: Sigma algebra, Random variables etc. (approximately Chap 1. of Durett)
Information Theory: Entropy, Mutual Information, etc.

Important Announcement
Exam Schedule

Midterm I will be held on Sep 26th in class.
Midterm II will be held on Oct 31st in class.
Midterm III will be held on Dec 3rd in class.
Each exam will last 1 hr.
1 letter sized cheat sheat (both sides) allowed. Closed Book, closed notes.
There will be no final exam.

List of Topics covered

Week Topic Additional Reading Assignment

Aug 18 - Aug 24

Introduction; what to expect in the course

Aug 25 - Aug 31

Proof of the Fundamental Theorem of Arithmetic (unique factorization)
Proof of infintely many primes
Twin primes conjecture
Johnson Lindenstrauss lemma
Robin's Theorem about the Riemann Hypothesis
P versus NP

Paper on the Riemann Hypothesis
Paper on the Johnson Lindenstrauss lemma

Sep 01 - Sep 07

Peano Axioms
Natural numbers, Integers
Abstract formulation of Rational Numbers
Proof of Pythogorus Theorem
Sqrt(2) is not rational
Partial and Total order
Least upper bound (Supremum) and Greatest lower bound (infimum)

Peano Axioms
Zermelo Frankel Set Theory

Sep 08 - Sep 14

Least upper bound property.
Proof: Rationals do not have the LUB property
Theorem: LUB property IFF GLB property.
Reals as Dedekind cuts
Field axioms
Fields: The rational and real fields
Functions; injective, surjective and bijective
Cardinality of Integers, Rationals and Reals
Countably infinite (proof for rationals)

More on Dedekind Cuts
We will not prove completeness but you can read it here.

Sep 15 - Sep 21

Reals are uncountable (Cantor's diagonalization)
Cardinality of set versus Power set (Cantor's diagonalization)
Aleph notation.
The continuum hypothesis
Convergent sequence, Cauchy sequence
Theorem: Every convergent sequence is a Cauchy sequence
Metric spaces

Sep 22 - Sep 28

Metric spaces continued
epsilon neighborhoods, limit points, interior points
Open sets, Closed sets
Thm: Open iff complement is closed
Thm: epsilon neighborhood is open
MIDTERM-1

Sep 29 - Oct 05

Complete Metric space; Seperable Metric Space
Thm: Arbitrary union, Finite intersection of open sets is open
Thm: Arbitrary intersection, Finite union of closed sets is closed
The L_p metric.
Definition: Topological space.
Definition: Topological space, trivial topology, discrete topology

Oct 06 - Oct 12

Compact sets; properties and theorems.
Thm: compact set is closed.
Stmt of Heine–Borel theorem
Continuity: Limit definition of continuous functions
Weierstrass's definition of continuous fns
Proof of equivalance
Pointwise continuity, Uniform continuity
Compactness and continuity
Proof: Continuous functions map compact sets to compact sets.

Oct 13 - Oct 19

C[a,b] as a metric space
Uniform convergence and Pointwise convergence of functions
Lagrange Polynomials and interpolation

Uniform limit theorem
No metric for pointwise convergence

Oct 20 - Oct 26

Bernstein Polynomials; properties.
Weierstrass Approximation theorem

Proof of the WAT
Note that there are typos on page 4. Should be K2 instead of K1 in the first two inequalities.

Oct 27 - Nov 02

Finished Weierstrass Approximation theorem
Linear/Vector spaces; examples
Definition of Vector/Linear space on a Field.
Subspace, how to prove
Span of a set of vectors (non-constructive/constructive definition)
Midterm-2

Nov 03 - Nov 09

Linear independence, Basis
Examples: Lagrange polynomial basis
Dimension of a vector space; proof using Steinitz lemma
Linear maps; examples
FFT and alternationg between different representations of polynomials
Matrix representation of a linear map

Steinitz exchange lemma

Nov 10 - Nov 16

Change of basis as a linear transform
Worked out example of Lagrange Polynomials as a basis
Composition of linear map and matrix matrix multiplication.
Rank Nullity Theorem, proof
Inverse of a linear map and matrix inverse
Ax=b, solution via Gaussian elimination.
LU decomposition
Normed (+Complete=Banach) vector spaces.
Lp Norms

Nov 17 - Nov 23

Inner product spaces: when field is real and is complex
Inner product (+Complete=Hilbert) spaces
Induced norm.
Quadratic form; symmetric and Hermitian matrix
Projection of a vector onto another
Pythogorean theorem
Cauchy Schwartz inequality, proof
Triangular inequality, proof.
Orthonormal basis; advantages.
Gram Schmidt orthogonalization and orthonormalization
QR Decomposition
Determinant of a matrix and signed volume

Nov 24 - Nov 30

Thanksgiving break

Dec 01 - Dec 07

Fourier Series; Riesz-Fiacsher theorem
Midterm-3

Week	Topic	Additional Reading	Assignment
Aug 18 - Aug 24	Introduction; what to expect in the course
Aug 25 - Aug 31	Proof of the Fundamental Theorem of Arithmetic (unique factorization) Proof of infintely many primes Twin primes conjecture Johnson Lindenstrauss lemma Robin's Theorem about the Riemann Hypothesis P versus NP	Paper on the Riemann Hypothesis Paper on the Johnson Lindenstrauss lemma
Sep 01 - Sep 07	Peano Axioms Natural numbers, Integers Abstract formulation of Rational Numbers Proof of Pythogorus Theorem Sqrt(2) is not rational Partial and Total order Least upper bound (Supremum) and Greatest lower bound (infimum)	Peano Axioms Zermelo Frankel Set Theory
Sep 08 - Sep 14	Least upper bound property. Proof: Rationals do not have the LUB property Theorem: LUB property IFF GLB property. Reals as Dedekind cuts Field axioms Fields: The rational and real fields Functions; injective, surjective and bijective Cardinality of Integers, Rationals and Reals Countably infinite (proof for rationals)	More on Dedekind Cuts We will not prove completeness but you can read it here.
Sep 15 - Sep 21	Reals are uncountable (Cantor's diagonalization) Cardinality of set versus Power set (Cantor's diagonalization) Aleph notation. The continuum hypothesis Convergent sequence, Cauchy sequence Theorem: Every convergent sequence is a Cauchy sequence Metric spaces
Sep 22 - Sep 28	Metric spaces continued epsilon neighborhoods, limit points, interior points Open sets, Closed sets Thm: Open iff complement is closed Thm: epsilon neighborhood is open MIDTERM-1
Sep 29 - Oct 05	Complete Metric space; Seperable Metric Space Thm: Arbitrary union, Finite intersection of open sets is open Thm: Arbitrary intersection, Finite union of closed sets is closed The L_p metric. Definition: Topological space. Definition: Topological space, trivial topology, discrete topology
Oct 06 - Oct 12	Compact sets; properties and theorems. Thm: compact set is closed. Stmt of Heine–Borel theorem Continuity: Limit definition of continuous functions Weierstrass's definition of continuous fns Proof of equivalance Pointwise continuity, Uniform continuity Compactness and continuity Proof: Continuous functions map compact sets to compact sets.
Oct 13 - Oct 19	C[a,b] as a metric space Uniform convergence and Pointwise convergence of functions Lagrange Polynomials and interpolation	Uniform limit theorem No metric for pointwise convergence
Oct 20 - Oct 26	Bernstein Polynomials; properties. Weierstrass Approximation theorem	Proof of the WAT Note that there are typos on page 4. Should be K2 instead of K1 in the first two inequalities.
Oct 27 - Nov 02	Finished Weierstrass Approximation theorem Linear/Vector spaces; examples Definition of Vector/Linear space on a Field. Subspace, how to prove Span of a set of vectors (non-constructive/constructive definition) Midterm-2
Nov 03 - Nov 09	Linear independence, Basis Examples: Lagrange polynomial basis Dimension of a vector space; proof using Steinitz lemma Linear maps; examples FFT and alternationg between different representations of polynomials Matrix representation of a linear map	Steinitz exchange lemma
Nov 10 - Nov 16	Change of basis as a linear transform Worked out example of Lagrange Polynomials as a basis Composition of linear map and matrix matrix multiplication. Rank Nullity Theorem, proof Inverse of a linear map and matrix inverse Ax=b, solution via Gaussian elimination. LU decomposition Normed (+Complete=Banach) vector spaces. Lp Norms
Nov 17 - Nov 23	Inner product spaces: when field is real and is complex Inner product (+Complete=Hilbert) spaces Induced norm. Quadratic form; symmetric and Hermitian matrix Projection of a vector onto another Pythogorean theorem Cauchy Schwartz inequality, proof Triangular inequality, proof. Orthonormal basis; advantages. Gram Schmidt orthogonalization and orthonormalization QR Decomposition Determinant of a matrix and signed volume
Nov 24 - Nov 30	Thanksgiving break
Dec 01 - Dec 07	Fourier Series; Riesz-Fiacsher theorem Midterm-3