Linear Algebra II

April, 2021

A 15 lecture course which introduces intermediate level topics in Linear Algebra.

Prerequisites

Assumes that Linear Algebra I is covered. Specifically assumes knowledge of the following topics:

  • Solutions to Linear Equations, Echelon Forms, Gaussian Elimination
  • Field, Vector Space Defintions, Real, Complex, Finite Field based Vector Spaces
  • Linear Transformation, Matrix, Rank, Inverse, Transpose
  • Change of Basis and effect on Matrix of the Linear Transformation
  • Determinants

Schedule

Meetings

Live Lectures

From 30th March to 4th May, 2021, Tuesdays, Thursdays and Saturdays

  • 10-11 AM for Batch 2
  • 12-01 PM for Batch 1
Tutorials

Weekly tutorials conducted by TAs in smaller groups as per times fixed.

Office Hours

I will be available on Teams during the timings given bellow, for clearing any doubts. You can sent a direct message to me for joining.

  • Monday, 530-730PM.
  • Thursday, 330-530PM.

Expected Workload

Students are expected to spent atleast 12 hrs per week. Roughly

  • 3+1 hrs attending lectures and tutorials
  • 4 hrs reading textbooks, references etc
  • 4 hrs solving assignments, quizes etc

Evaluations

  • 4 Light Quizzes (Weekly)
  • 3 Assignments
  • 2 Deep Quizzes (17th April, 4th May)

Textbook and References

Textbook

The resources provided are licenced under Creative Commons/Open Licences and hence downloadable.

Extra Reading

On Solving Problems & Understanding Proofs

Lecture Topics

Eigenvalues & Diagonalization

1. Linear Algebra & Random Walks

Recalling Basics | Function Spaces | Random Walk on Graphs

2. Eigenvectors and Eigenvalues

Definitions | Characteristic Polynomial | Examples

  • Notes | Video
  • Reading
    • Section 3.3 for Eigenvalues, Section 2.9 for Random Walks on Graphs in [WKN].
    • Chapter 12 in [CDTW].

3. Diagonalization

Eigenvector Basis and Powering | Multiplicities

  • Notes | Video
  • Reading
    • Section 3.3 in [WKN]
    • Chapter 13 in [CDTW]

Assignment 1

Submit by 13th April

Norms & Inner Products

4. Norms & Inner Products

Jordan Form | Norms | Distance | Inner Product | Complex Case

5. Orthonormal Vectors

Orthogonal & Orthonormal Vectors | Gram-Schmidt Orthogonalization

6. Projection and Orthogonal Complement

Subspace Projections | Orthogonal Complements | Fitting with Errors

  • Notes | Video
  • Reading
    • Section 14.6 in [CDTW]
    • Section 8.1 in [KTW]

Advanced Topics

7. Least Squares Fitting

Best fit vector on a subspace | Least Squares Fitting Equation

8. Deep Quiz I and Discussion

Assignment 2

Submit by 26th April

9. Spectral Decomposition Theorem

10. Singular Value Decomposition

11. Problem Solving

Applications

12. PCA & Best Fit Subspaces

Assignment 3

Submit by 27nd April

13. Sparse Recovery and Pooled Testing (Tentative)

14. Course Summary & Closing Notes.

15. Deep Quiz II

On 4th May