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Algebra 1: Foundations

NPTEL via Swayam

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Details

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Provider

Swayam
Pricing

Free Online Course
Languages

English
Certificate

Paid Certificate Available
Effort

12 weeks
Sessions

Self-Paced
Level

Intermediate

Found in

ABOUT THE COURSE: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 engineeringPREREQUISITES: High school mathematicsINDUSTRY SUPPORT: Microsoft, Google, etc.

Syllabus

Week 1: Sets and Logic
Week 2:Equivalence relations, partitions and the quotient construction on sets, functions.
Week 3:Countability, Isomorphisms of sets and the canonical decomposition of maps between sets.
Week 4:Binary operations, the definition, examples and basic properties of groups, rings and integral domains
Week 5:The natural numbers.
Week 6:Fields, the rational, real and complex numbers, polynomials and rational functions
Week 7:System of linear equations, Gaussian elimination, Matrices
Week 8:Vector spaces, examples of vector spaces, subspaces, spans, linear independence, bases and dimension.
Week 9:Linear maps, the rank—nullity theorem and applications.
Week 10:The connection between linear maps and matrices. Change of basis. Dual spaces, quotient spaces.
Week 11:The determinant as a multilinear map. Determinants and volumes, linear operators, eigenvectors and the characteristic polynomial
Week 12:Triangular and diagonal forms, the Jordan form