Linear Algebra II

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.

• [CDTW] Linear Algebra
  David Cherney, Tom Denton, Rohit Thomas and Andrew Waldron

• [WKN] Linear Algebra with Applications
  W. Keith Nicholson

• [MR] Interative Linear Algebra
  Dan Margalit and Joseph Rabinoff

Extra Reading

• Down with Determinants
  Sheldon Axler

• Linear Algebra Done Right Videos
  Sheldon Axler

• The Fast Fourier Transform and Polynomial Multiplication
  Mordecai GOLIN

• [GS] Linear Algebra MIT OCW Course Materials
  Gilbert Strang

On Solving Problems & Understanding Proofs

• Richard Feynman, His Life, and the Art of Solving Important Problems
  Blog Post

• How to Solve it
  G. Polya
  Shorter Version

Lecture Topics

Eigenvalues & Diagonalization

1. Linear Algebra & Random Walks

Recalling Basics | Function Spaces | Random Walk on Graphs

• Notes | Video
• Reading
  • Section 2.9 in [WKN]

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

• Notes | Video
• Reading
  • Section 10.1 in [WKN]
• Explore
  • For Jordan Form, Generalized Eigenvectors, see Down with Determinants.
  • Geometric Multiplicity $\leq$ Algebraic Multiplicity.
  • Rotation Matrics have complex eigenvalues.

5. Orthonormal Vectors

Orthogonal & Orthonormal Vectors | Gram-Schmidt Orthogonalization

• Notes | Video
• Reading
  • Section 10.2 in [WKN]
  • Chapter 14 in [CDTW]
• Explore
  • For Bilinear forms see Slides of Prof. P. Karageorgis.

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

• Notes | Video
• Reading
  • [MR] Section 6.5
  • Chapter 7 (upto page 307) in [CDTW]

8. Deep Quiz I and Discussion

• Notes | Video

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)

• Explore
  • Compressive Sensing
    Lecture Notes. Tim Roughgarden & Gregory Valiant.

14. Course Summary & Closing Notes.

15. Deep Quiz II

On 4th May