Online platform: MS Teams (email me for access)
Lectures
 Linear system of equations


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

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

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

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

 Vector spaces


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

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

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

07 Feb: Four fundamental subspaces in linear algebra, onesided matrix inverses notes video

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

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

 Orthogonality


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

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

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

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

28 Feb: Orthogonal matrices and their properties notes video

01 Mar: Tall orthogonal matrices, GramSchmidt orthogonalization process, QR decomposition notes video

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

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

 Determinants
 Eigenvalue problems


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

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

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

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

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

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

 Positive Definite Matrices and the Singular Value Decompositiom


04 Apr: Optimization viewpoint motivation of quadratic forms (links of previous lecture)

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

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

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

25 Apr: Summary lecture.

Tutorials

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

Vector spaces on 17 Feb. Quiz on 24 Feb.

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

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

Eigenvalue problems on 12 Apr. Quiz on 21 Apr.

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


Group formation, 28 March

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

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

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

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 6070% picks out the linear algebra aspects and explains them, final 1520% 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


Tutorial quizzes (approx 4) and tutorial participation

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

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 503020 range for the three items above.
Outline
 Broad course contents


Linear system of equations

Vector spaces

Orthogonality

Eigenvalue problems

Positive definite matrices

Singular value decomposition

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


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

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