## 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.
• Extra problems.

Week 6 :

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 :

Week 12 :

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.