Vector Space Linear Algebra

ISSS Certificate of Teaching

Motivation

In this self-designed course, I teach Linear Algebra from a more abstract point of view i.e., using Vector Spaces. The focus is on building the required mathematical rigour to think in a more abstract way and then connect to more tangible concepts like Matrices.

Experience

I realised how much goes into building a course that is self-sufficient, engaging and such that the students feel they have learned something useful. I definitely get our professors much better now! Handling diverse doubts from students helped me also learn and strengthen my concepts. Since it was live and interactive, a teacher has to think on the spot and even have the humility to say that they don’t know something when they don’t know it.

Course Design & Content

I designed and executed the course under the PMRF Student Lecture Series scheme of ISSS, which provides a platform to connect with interested students. The Course Flyer had the schedule and the planned course content. All the lectures were held online in Google Meet which allowed anyone around the world to attend. Every lecture was for two hours. All the lectures were recorded and uploaded in ISSS Course Website.

All the lectures are uploaded in the Vector Space Linear Algebra Course playlist of my YouTube teaching channel.

• Lecture 1: Motivation & Basics

  • Why do we need Vector Spaces?
  • Defining & Discussing - Abelian Groups, Fields & Vector Spaces

• Lecture 2: Subspaces

  • Problems on Abelian Groups, Fields and Vector Spaces
  • Define & Discuss Subspaces

• Lecture 3: Basis & Dimensions and Linear Transformations

  • Introduce Basis & Dimensions and Linear Transformations.
  • Define some important subspaces - Range and Null Space

• Lecture 4: Algebra of Linear Transformations, Eigen, Coordinates

  • Introducing Algebra of Linear Transformations, Eigen Vectors & Eigen Values.
  • Theorems on Bases & Dimensions
  • Introducing Coordinates!

• Lecture 5: Problems from Hoffman-Kunze

  • Problems on Subspaces, Basis & Dimensions and Coordinates
  • Rank-Nullity Theorem proof
  • Problems on Linear Transformations & Algebra of Linear Transformations

• Lecture 6: Isomorphism, Representation of Transformations & First Matrix View

  • Problems on Algebra of Linear Transformations
  • Introduction & Problems: Isomorphisms, Representation of Transformations
  • Matrix Views: 1. System of Linear Equations

• Lecture 7: Matrices, Matrices, Matrices!

  • Important theorem of Eigenvalues and Eigenvectors
  • Rank & Invertibility for Matrices
  • Two more views of Matrices
  • PSD and PD Matrices