Algebra & Algorithms

  • 0.0
Approx. 30 hours to complete

Course Summary

Algebra and Algorithms is an online course that teaches how to use algebraic reasoning to solve problems, and how to design algorithms that exploit algebraic structure. The course covers topics such as polynomial equations, matrices, and linear algebra.

Key Learning Points

  • Learn how to use algebraic reasoning to solve problems
  • Discover how to design algorithms that exploit algebraic structure
  • Understand polynomial equations, matrices, and linear algebra

Job Positions & Salaries of people who have taken this course might have

  • Data Scientist
    • USA: $113,309
    • India: ₹1,130,000
    • Spain: €36,000
  • Software Engineer
    • USA: $105,563
    • India: ₹789,000
    • Spain: €30,000
  • Financial Analyst
    • USA: $64,332
    • India: ₹580,000
    • Spain: €24,000

Related Topics for further study


Learning Outcomes

  • Master algebraic reasoning and algorithm design
  • Develop a strong understanding of polynomial equations, matrices, and linear algebra
  • Apply algebraic reasoning and algorithms to real-world problems

Prerequisites or good to have knowledge before taking this course

  • Basic knowledge of algebra and calculus
  • Familiarity with programming concepts

Course Difficulty Level

Intermediate

Course Format

  • Online
  • Self-paced

Similar Courses

  • Linear Algebra for Beginners: Open Doors to Great Careers
  • Introduction to Algebra
  • Mathematics for Machine Learning: Linear Algebra

Related Education Paths


Notable People in This Field

  • Gilbert Strang
  • Terence Tao

Related Books

Description

Algebra is one of the definitive and oldest branches of mathematics, and design of computer algorithms is one of the youngest. Despite this generation gap, the two disciplines beautifully interweave. Firstly, modern computers would be somewhat useless if they were not able to carry out arithmetic and algebraic computations efficiently, so we need to think on dedicated, sometimes rather sophisticated algorithms for these operations. Secondly, algebraic structures and theorems can help develop algorithms for things having [at first glance] nothing to do with algebra, e.g. graph algorithms. One of the main goals of the offered course is thus providing the learners with the examples of the above mentioned situations. We believe the course to contain much material of interest to both CS and Math oriented students. The course is supported by programming assignments.

Knowledge

  • Efficiently doing arithmetics on binary numbers as well as algebraic operations like polynomial multiplication, matrix multiplication and inversion.
  • Design efficient algorithms problems in graph theory related to distances and matchings based on fast matrix computations and randomization.

Outline

  • Arithmetics in the Realm of Circuits
  • Course Trailer
  • Welcome to the Course
  • Definition of Boolean circuits
  • Circuit implementation of (Naïve) Integer Addition
  • Parallel “Product” Prefix computation
  • Applying Parallel Prefix to improve Adder depth
  • Circuit for Subtraction of Integers
  • Circuit for Integer Multiplication
  • Circuit for Integer Division
  • Circuit for Integer Division (continued)
  • Soft and Hard Prerequisites for the Course
  • Module Prerequisites
  • Boolean Circuits for Arbitrary Functions
  • Lower bound on the circuit size: from circuits to trees
  • Lower bound on the circuit size (continued)
  • Circuits for arbitrary functions: Ingredient 1 — Shannon Expansion
  • Circuits for arbitrary functions: Ingredient 2 — Decoder
  • Circuits for arbitrary functions: Ingredient 3 — Universal Multipole
  • Circuits for arbitrary functions: Conclusion
  • Module Prerequisites
  • More on Multiplication of Integers and Polynomials
  • Karatsuba—Ofman Multiplication in subquadratic time
  • Strassen’s matrix multiplication algorithm
  • Toom’s algorithm
  • Toom’s algorithm (continued)
  • Fast Fourier Transform: efficient polynomial evaluation and interpolation
  • Fast Fourier Transform (continued)
  • Fast Fourier Transform (conclusion)
  • Module Prerequisites
  • Graph Reachability and Distances via Matrix Multiplication
  • Matrix multiplication exponent
  • Computing Transitive Closure of a Graph
  • Computing Distances: Seidel’s algorithm
  • Seidel’s algorithm (continued)
  • Boolean Matrix Product: Four Russians Speedup Trick
  • Four Russians Speedup Trick (continued)
  • Module Prerequisites
  • More on Matrix Computations
  • Computing determinants in parallel: Chistov’s Algorithm
  • Chistov’s Algorithm (continued)
  • Computational equivalence of matrix operations: Inversion to Multiplication
  • Computational equivalence of matrix operations: Multiplication to Inversion
  • LUP Matrix Decomposition
  • Module Prerequisites
  • Matchings in Graphs via Matrix Determinants
  • Back to Roots: DeMillo–Lipton–Schwarz–Zippel lemma
  • First Application of DLSZ Lemma: Checking Matrix Multiplication
  • Edmonds’ Matrices and Matchings in Bipartite Graphs
  • Tutte’s Matrices and Matchings in Arbitrary Graphs
  • Uniqueness of Optimization Problem Solutions: the Isolating Lemma
  • From Checking for Existence to Finding a Perfect Matching
  • Module Prerequisites

Summary of User Reviews

A highly rated course on Coursera that teaches algebra and algorithms. Many users appreciated the clear explanations of complex concepts.

Key Aspect Users Liked About This Course

Clear explanations of complex concepts

Pros from User Reviews

  • Well-structured course content
  • Engaging and knowledgeable instructors
  • Challenging assignments that reinforce learning
  • Great for both beginners and advanced learners
  • Flexible pacing and schedule

Cons from User Reviews

  • Some users found the pace too slow
  • Lack of interaction with other students
  • Limited opportunity for practice
  • Minimal feedback on assignments
  • Requires a basic understanding of math concepts
English
Available now
Approx. 30 hours to complete
Дмитрий Ильинский Top Instructor, Alex Dainiak
Moscow Institute of Physics and Technology
Coursera
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