## MATH/MTHE 212: Linear Algebra II

Winter 2018

### Learning Resources

**An abridged version of the course textbook is available here.**This is a freely available document that can be used for

**suggested reading**. However, all examples, exercises and most comments from the original text have been removed but with chapters, sections, definitions and results etc. numbered as in the original text. I shall use the numbering from the textbook and you should be able to use the above document to understand what is being referred to.

**Week 1 :**

- Topics: Vector spaces, subspaces, sums and direct sums.
- Suggested reading: Chapter 1 of the book.
- Summary of Week 1.
- No tutorial in Week 1 .

**Week 2 :**

- Topics: Span, Linear Independence and Basis of a Vector Space.
- Suggested reading: Chapter 2, Sections 2.A and 2.B of the book
- Summary of Week 2 which contains details of proofs that were skipped/sketched in class.
- Week 2
**Tutorial**.

**Week 3 :**

- Topics: Dimension, Linear maps, Kernel and Image of linear maps
- Suggested reading: Chapter 2- Sections 2.C and Chapter 3 - Section 3.A and 3.B of the book.
- Summary of Week 3.
- Week 3 Tutorial.

**Week 4 :**

- Topics: Matrix of a linear map, Invertible linear maps, Invariant subspaces.
- Suggested reading: Chapter 3- Sections 3.C and 3.D, Chapter 5- Section 5.A of the book.
- Summary of Week 4.
- Week 4 Tutorial.

**Week 5:**

- Topics: Upper triangular matrices, Diagonal matrices, Inner Product spaces.
- Suggested reading: Chapter 5- Sections 5.B and 5.C, Chapter 6- Section 6.A of the book.
- Summary of Week 5. Click here for a write-up illustrating the application done in Wednesday's class.
- Week 5 Tutorial.
- A checklist to help you prepare for the midterm.
- Extra problems.

**Week 6 :**

- Topics:Norms, orthonormal vectors, Gram-Schmidt process.
- Suggested reading: Chapter 6- Sections 6.A and (the start of) 6.B
- Summary of Week 6.
- Week 6 Tutorial.
- Midterm with solutions.

**Week 7 :**

- Topics:Orthogonal projections, minimizing distance, method of least squares, adjoints.
- Suggested reading: Chapter 6- Section 6.B and Chapter 7: (the start of) 7.A, notes on least squares solutions given in the summary notes.
- Summary of Week 7.
- Week 7 Tutorial.

**Week 8 :**

- Topics:Self-adjoint operators, Normal operators, Spectral theorems, Positive operators, Isometries.
- Suggested reading: Chapter 7- Secttion 7.A, 7.B (only statements of the two spectral theorems) and 7.C.
- Summary of Week 8.
- Week 8 Tutorial.

**Week 9 :**

- Topics: Finding square roots of positive operators, Polar Decomposition and Singular Value Decomposition.
- Suggested reading: Chapter 7- Secttion 7.D and the summary containing detailed notes on SVD
- Summary of Week 9.
- Week 9 Tutorial.

**Week 10 :**

- Topics: Generalized eigenvectors, multiplicity of an eigenvalue, Characteristic and minimal polynomials
- Suggested reading: Chapter 8- Secttions 8.A ,8.B, 8.C
- Summary of Week 10.
- Week 10 Tutorial.

**Week 11 :**

- Topics: Decomposition of operators (contd.), Jordan Forms
- Suggested reading: The summary file below.
- Summary of Week 11.
- Week 11 Tutorial.

**Week 12 :**

- Topics: Jordan Forms problems, applications of SVD (won't be tested) and Review.
- Suggested reading: The summary file below.
- Summary of Week 12.
- Week 12 Tutorial.
- See here and here for material used for the special lecture.
- Write-up for Polar decomposition using SVD.

**Exam preparation :**- A checklist to help you prepare for the exam, especially to have an idea of the level of proof-based problems. It also includes a list of learning outcomes.
- Extra Problems. Please look at the Solutions only after trying seriously to solve the problems.

© 2017 Neha Prabhu