Week 1 :

• Topics: Vector spaces, subspaces, sums and direct sums.

• Summary of Week 1.

• No tutorial in Week 1 .

Week 2 :

• Topics: Span, Linear Independence and Basis of a Vector Space.

• 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

• Summary of Week 3.

• Week 3 Tutorial.

Week 4 :

• Topics: Matrix of a linear map, Invertible linear maps, Invariant subspaces.

• Summary of Week 4.

• Week 4 Tutorial.

Week 5:

• Topics: Upper triangular matrices, Diagonal matrices, Inner Product spaces.

• Summary of Week 5.

• Week 5 Tutorial.

• A checklist to help you prepare for the midterm.

• Extra problems.

Week 6 :

• Topics:Norms, orthonormal vectors, Gram-Schmidt process.

• Summary of Week 6.

• Week 6 Tutorial.

• Midterm with solutions.

Week 7 :

• Topics:Orthogonal projections, minimizing distance, method of least squares, adjoints.

• 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.

• Summary of Week 8.

• Week 8 Tutorial.

Week 9 :

• Topics: Finding square roots of positive operators, Polar Decomposition and Singular Value Decomposition.

• Summary of Week 9.

• Week 9 Tutorial.

Week 10 :

• Topics: Generalized eigenvectors, multiplicity of an eigenvalue, Characteristic and minimal polynomials

• Summary of Week 10.

• Week 10 Tutorial.

Week 11 :

• Topics: Decomposition of operators (contd.), Jordan Forms

• Summary of Week 11.

• Week 11 Tutorial.

Week 12 :

• Topics: Jordan Forms problems, applications of SVD (won't be tested) and Review.

• Summary of Week 12.

• Week 12 Tutorial.

• 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.