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Algebra 1: Foundations

Prof. Jaikrishnan J · IIT Palakkad

Start with lecture 1 At NPTEL

About

This is part 1 of a series of courses on Algebra at the undergraduate level. The first course is the foundational course. We will spend half of the course on fundamentals which are often skirted in college courses. The aim of the first six weeks is to have a thorough understanding of the basic objects that form the backbone of modern algebra. Along the way, we will also introduce the notion of proof in an elementary setting. The latter six weeks is an introduction to linear algebra culminating in an elementary proof of the existence of the Jordan form given by Terence Tao. Throughout the course we will sprinkle interesting tidbits about history.

Intended audience: Bachelor’s students in the sciences and engineering

Prerequisites: High school mathematics

Lectures

Week 1

  1. The nature of pure mathematics YouTube
  2. Basic set theory YouTube
  3. Basic logic YouTube
  4. Proofs YouTube
  5. More about proofs YouTube
  6. Set theoretic identities YouTube

Week 2

  1. Functions YouTube
  2. Equivalence relations YouTube
  3. Quotients YouTube
  4. More functions YouTube

Week 3

  1. Binary operations YouTube
  2. Groups YouTube
  3. Rings and integral domains YouTube
  4. Peeking under the carpet YouTube

Week 4

  1. The Integers YouTube
  2. Induction YouTube
  3. Applications of Induction. YouTube
  4. Uniqueness of Integers. YouTube

Week 5

  1. Cardinality of sets YouTube
  2. Division in the Integers YouTube

Week 6

  1. Congruences YouTube
  2. Basic number theory YouTube
  3. Fields YouTube

Week 7

  1. Real numbers YouTube
  2. Complex numbers YouTube
  3. Polynomials YouTube

Week 8

  1. System of Linear Equations YouTube
  2. Gaussian elimination YouTube
  3. Matrix Algebra YouTube
  4. Square systems. YouTube
  5. The set of solutions YouTube

Week 9

  1. Vector spaces YouTube
  2. Direct sums of subspaces YouTube
  3. Spans and linear independence YouTube
  4. Basis YouTube

Week 10

  1. Linear Maps YouTube
  2. The Rank--Nullity theorem YouTube
  3. Matrices and Linear transformations YouTube
  4. Change of basis YouTube

Week 11

  1. Eigenvalues YouTube
  2. Minimal Polynomial YouTube
  3. Applications of the minimal polynomial YouTube

Week 12

  1. Triangular Form YouTube
  2. Jordan Form YouTube
  3. Examples of Canonical Forms YouTube

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IIT Palakkad
Department
Mathematics
Teacher
Prof. Jaikrishnan J
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge. Coordinated by IIT Madras.
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