Online platform: MS Teams (email me for access)

Lectures

Linear system of equations

  1. 17 Jan: Introduction and row picture of a system of equations notes, video

  2. 18 Jan: Col picture of a matrix equation, refresher of Gaussian elimination and elementary matrices notes video

  3. 24 Jan: LU decomposition of a matrix, pivoting and numerical issues notes, video

  4. 25 Jan: Matrix inverses, constructing sparse matrices from PDEs — example using Poisson’s eqn notes video

Vector spaces

  1. 27 Jan: Introduction to vector spaces and an informal view of the four fundamental spaces of a matrix notes video

  2. 31 Jan: Echelon and row reduced echelon form, connection of the pivot/free variables with the col/null space of the matrix notes video

  3. 01 Feb: Linear independence of vectors, spanning set for a vector space, basis of a vector space notes video

  4. 07 Feb: Four fundamental subspaces in linear algebra, one-sided matrix inverses notes video

  5. 08 Feb: Linear transformations and how to express them as matrices notes video

  6. 10,14 Feb: Linear transformations (contd) with examples notes video

Orthogonality

  1. 14 Feb: Norm of a vector and linear independence of orthogonal vectors notes video

  2. 15 Feb: Orthogonality of subspaces, orthogonal compliments and examples, constructing linear models from data notes video

  3. 21 Feb: Solving an over determined system of equations, least squared error solutions notes video

  4. 22 Feb: Solving an under determined system of equations notes video

  5. 28 Feb: Orthogonal matrices and their properties notes video

  6. 01 Mar: Tall orthogonal matrices, Gram-Schmidt orthogonalization process, QR decomposition notes video

  7. 07 Mar: Hilbert and function spaces, connections with Fourier series notes video

  8. 08 Mar: Polynomial approximations in function spaces, orthogonal functions via Gram Schmidt notes video

Determinants

  1. 08 Mar: Fundamental properties of determinants (see notes/video above)

  2. 10 Mar: Derived properties of determinants, application to finding volumes of n-dim boxes notes video

Eigenvalue problems

  1. 14 Mar: Definitions and some properties of eigenvalue problems notes link: video

  2. 15 Mar: Geometric and algebraic multiplicities of eigenvalues, linear independence of eigenvectors with different eigenvalues notes video Additional notes here and here.

  3. 21 Mar: Diagonalization of a matrix, powers of a matrix, Hemachandra (aka Fibonacchi) numbers via powers of a matrix notes video

  4. 22 Mar: Properties of Hermitian matrices, specialization to real valued matrices and the spectral theorem notes video

  5. 28 Mar: Spectral theorem, similarity transformations notes video (first 10 mins missing).

  6. 29 Mar: Similarity transforms and connections to change of variables and linear transformations notes video

  7. 04 Apr: Schur decomposition of a matrix notes video

Positive Definite Matrices and the Singular Value Decompositiom

  1. 04 Apr: Optimization view-point motivation of quadratic forms (links of previous lecture)

  2. 05 Apr: Quadratic forms and positive definite properties, tests for positive definite matrices notes video

  3. 11 Apr: The SVD and its proof notes, also see here video

  4. 18 Apr: SVD and the four fundamental spaces of a matrix, outer-product form of the SVD, example of truncated SVD with an example of image compression, code & input, notes video

  5. 19 Apr: SVD and matrix computations notes, extra on condition number, video, reference

  6. 25 Apr: Summary lecture.

Tutorials

  1. Linear system of equations on 03 Feb. Quiz on 14 Feb.

  2. Vector spaces on 17 Feb. Quiz on 24 Feb.

  3. Orthogonality - part 1 on 11 Mar. Quiz on 24 Mar.

  4. Orthogonality - part 2 with determinants on 31 Mar. Quiz on 07 Apr.

  5. Eigenvalue problems on 12 Apr. Quiz on 21 Apr.

  6. Positive Definite matrices and SVD on 26 Apr. No quiz.

Course Project

In teams of 2 (not 1, not 3), students will create YouTube videos about some aspect of Linear Algebra in under 10 minutes.

These are the stages of the project, and details must be entered into the shared spreadsheet

  1. Group formation, 28 March

  2. Fixing the broad area, 04 April, more details to follow here

  3. Fixing the title of the project and the main reference, 18 April (2pm)

  4. Submitting a one page summary, 25 April — this has 15% weightage. Sample template file — copy/paste into overleaf.

  5. Submitting final YouTube link, 17 May (2pm) — this has 85% weigthage. An approximate guideline for the video — Time budget for your video (approximate guidelines): first 15% lays out the problem at a "40,000 feet" view, next 60-70% picks out the linear algebra aspects and explains them, final 15-20% connects the linear algebra aspects back to the original problem and shows how the original problem is solved. Please identify the relevant linear algebra aspects very clearly.

Evaluation

The course evaluation consists of

  1. Tutorial quizzes (approx 4) and tutorial participation

  2. End semester exam — In person, closed notes but two sided A4 sheet written in one’s own handwriting allowed

  3. Course project (e.g. a 10 min video explaining some concepts)

The exact distribution for each component is subject to the evolution of Covid related restrictions, but roughly it will be in the 50-30-20 range for the three items above.

Outline

Broad course contents

  1. Linear system of equations

  2. Vector spaces

  3. Orthogonality

  4. Eigenvalue problems

  5. Positive definite matrices

  6. Singular value decomposition

Note: If you have done any linear algebra course previously, you are ineligible to take this course.

References books

  1. Linear Algebra and its applications, Gilbert Strang, 4th ed. Keyword GS

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