EE5120 Linear Algebra (July-Dec 2017-18), Instructor: Dr Uday Khankhoje
Lectures (C slot): Mo 10-11a, Tu 9-10a, We 8-9a, Fr 12-1p (tutorials). All in ESB 350.
News
  1. One A4 sized cheat sheet allowed for end sem exam.
  2. Tutorial quiz schedule (tentative) announced below.
  3. Tutorial 4 - Quiz on 18 Sept.
  4. Tutorial 3 - Quiz and Tutorial - 4 discussion on 11 Sept.
  5. Mid-semester exam on Wed 27 Sept from 7-9am (you may bring coffee to the exam!).
  6. Tutorial 2 - Quiz on 18 Aug. Per course policy, solutions to tutorial will be posted after the quiz.
  7. Please self enroll in Moodle if you haven't already (by 27 Aug). Get the enrollment key from me or the TAs.
  8. No class on 14 Aug.
  9. Tutorial 1 - Quiz on 11 Aug
Tutorial dates:
Tutorial # 1, sol, Q 2, sol, Q 3, sol, Q 4, sol, Q 5, sol 6, sol, Q 7, sol, Q 8, sol, Q
Date 11 Aug 21 Aug 01 Sep 11 Sep 22 Sep 9 Oct 23 Oct 3 Nov
Quiz avg 6.0/10 5.4/10 6.3/10 5.9/10 - 8.3/10 3.2/10 5.7/10

Resources
  1. Gilbert Strang's website.
  2. A popular YouTube channel for visualizations in linear algebra.
  3. On using the open source software SAGE to do linear algebra: tutorial.
Lecture Topics
  1. Introduction and solving a linear system of equations, Ax=b (Ch 1 of GS) [5 lectures]
    1. Geometric (row) and algebraic (column) picture of matrix equations. Lecture 1, 31 Jul
    2. Refresher of Gaussian elimination. Lecture 2, 01 Aug
    3. Gaussian elimination as matrix multiplications: LU decomposition (sample code). Lecture 3, 02 Aug
    4. Pivoting, round-off errors, matrix inverse and transpose. Lecture 4, 04 Aug
    5. Finite difference matrices: tridiagonal and their LU decomposition. Lecture 5, 07 Aug
  2. Vector spaces (Ch 2 of GS) [11 lectures]
    1. Definitions of vector spaces and sub-spaces, column space of a matrix. Lecture 6, 08 Aug
    2. Column and null space of a matrix with examples. Lecture 7, 09 Aug
    3. Echelon and row reduced echelon form of a matrix, matrix rank and dimensionality of col space and null space. Lectures 8,9, 16,18 Aug
    4. Span of a vector space, basis, dimension, four fundamental subspaces related to a matrix. Lectures 10,11,12, 21,22,23 Aug
    5. Inverses of rectangular matrices. Lecture 13, 28 Aug
    6. Linear transformations. Why are matrix computations preferred? Discussion here. Lectures 14,15,16, 29,30 Aug, 04 Sept
    7. Vector norms, and a brief visit to non-Euclidean geometry. Lecture 16, 04 Sept
  3. Orthogonality (Ch 3 of GS) [7 lectures]
    1. Orthogonality of vectors, subspaces, notion of orthogonal compliment of a subspace, and orthogonality relations between the four fundamental subspaces of a matrix. Lecture 17, 05 Sept
    2. Solutions to least square error problems, and connection to pseudo-inverse. Lectures 18,19, 06 Sep, 11 Sept
    3. Projection onto a vector space as a matrix operation, projection onto a line. Minimum norm solution in the under-determined case, and connection to pseudo-inverse. Lecture 20, 12 Sept
    4. Orthogonal vector and matrices, Gram-Schmidt process of orthonormalization. Lecture 21 13 Sept
    5. QR decomposition of a matrix, Hilbert spaces, function spaces and the concept of orthogonality in these spaces. Lecture 22,23 15,18 Sept
  4. Determinants in brief (Ch 4 of GS) [2 lectures]
    1. Properties of determinants (sec 4.2 of GS). Lecture 24 19 Sept
    2. Geometrical interpretation of determinants; determinant of the Jacobian. Lecture 25 20 Sept
  5. Eigenvalues and eigenvectors (Ch 5 of GS) [10 lectures]
    1. Definition and a few properties of the matrix eigenvalue problem. Lecture 26 25 Sept
    2. Algebraic and geometric multiplicity of an eigenvalue, some properties; Proof regarding multiplicity. Lecture 27 03 Oct
    3. Diagonalization of a matrix (also called its eigen decomposition); its use to compute powers of a matrix. Note: Related to eigen decomposition is another matrix decomposition, the Schur decomposition. This is useful for matrices that are not diagonalizable, and can be proved by extending the above proof on multiplicity. Lecture 28 04 Oct
    4. Cayley-Hamilton theorem (two proofs). Lecture 29 06 Oct
    5. Powers of a matrix: application based on Fibonacci numbers, Hermitian matrices and their properties, Spectral theorem. Lecture 30 10 Oct
    6. Unitary matrices, change of basis and similarity transforms. Lecture 31,32 11,13 Oct
    7. On the Schur decomposition of a matrix (notes). Lectures 33,34,35 16,17,20 Oct
  6. Positive definite matrices and the SVD (Ch 6 of GS) [7 lectures]
    1. Idea of optimization, quadratic forms and definition of positive definite matrices, geometric interpretations. Lecture 36 24 Oct
    2. Tests for positive definiteness of matrices; Cholesky decomposition of a symmetric matrix. Lecture 37 25 Oct
    3. Singular value decomposition; its proof and various properties (links). Lectures 38,39 30,31 oct
    4. Applications of the SVD: image compression (sample code, image, output). Lecture 40 01 Nov
    5. SVD and matrix computations; psuedo-inverses, condition number, regularization (truncated SVD and Tikhonov). Lectures 41,42 6,7 Nov

Course Project
  1. Deadlines: Group formation (25 Sept), Title formation (13th Oct), Video submission (12 Nov). View only link to all projects/links/marks/comments.
  2. Guidelines and tips
    • 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.
    • Where to look for possible topics? Applications of linear algebra (url1, url2). Many IEEE societies publish magazines which explain topics at a high-level. These can be good starting points. Examples: IEEE Signal Processing Magazine, IEEE Communications Magazine, IEEE Antennas and Propagation Magazine.
    • Some areas that extensively use linear algebra: compressive sensing, image and signal processing, quantum computing, computer graphics, graphs and networks, numerical linear algebra.
    • Your choice of topic must be approved by the instructor (see spreadsheet)
    • Technical details of the video that you will make: duration 4-6minutes, and no robot voices.
    • Some tips on how to make a video: url1, url2.
    • Use http://goo.gl to shorten links.
Course flyer
  • Evaluation: 30% midsem, 25% tutorial quizs (best n out of n+1), 10% research paper summary, 35% endsem
  • Refer to the academic course listing for syllabus. In short, we will study most of the topics in the textbook, with an inclination towards numerical linear algebra where necessary.
  • Text book: Linear Algebra and its applications, Gilbert Strang, 4th ed. GS.

Policies
  • As per institute rules, 85% attendance (minimum) is mandatory and will be enforced.
  • Academic misconduct: There will be zero tolerance towards any unethical means, such as plagiarism (COPYING in plain and simple terms) or proxy attendance. Read these links to familiarize yourself, there will be no excuse for ignorance: URL1, URL2. Penalties incude: receiving a zero in a particular assignment/examination, receiving a fail grade for the entire course, having a note placed in your permanent academic record, suspension, or all of the above.


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