Lectures

Linear system of equations

  1. Introduction, row and column pictures of a linear system of equations. 27 July

  2. Refresher of Gaussian elimination and elementary matrices, LU decomposition of a matrix. 28 July

  3. Pivoting, partial pivoting, and numerical issues, Matrix inverses and their properties. 04 Aug

Additional resources: Notes on Linear system of equations, and related Matlab resources

Reading material: Chapter 1 of Gilbert Strang’s Linear Algebra

Vector spaces

  1. Qualitative overview of the four fundamental vector spaces of linear algebra. 06 Aug

  2. Row echelon and row-reduced echelon form as a way of scoping the null space of a matrix, formal definitions of vector spaces and sub-spaces. 10 Aug

  3. Solvability conditions on a system of linear equations, linear independence of vectors, vectors spanning a vector space, basis vectors for a vector space. 11 Aug

  4. Four fundamental vector spaces of a matrix (column, null, row, left null), relations between them, idea of matrix inverses for rectangular matrices. 17 Aug

  5. Linear transformations and matrices, examples via integration and differentiation operations. 18 Aug

  6. More examples of linear transformations (rotations, projections, etc.), Lp norm of a vector. 24 Aug

Additional resources: Notes on Vector spaces, and related Matlab resources

Reading material: Chapter 2 of Gilbert Strang’s Linear Algebra, except section 2.5 on graphs and networks

Orthogonality

  1. Basic definitions of orthogonality, orthogonality between vector spaces, orthogonal complements of vector spaces, introduction to the least squares problem. 25 Aug

  2. Solving an overdetermined system of equations, least squares error solutions. 01 Sept

  3. Projection operator, solving an overdetermined system of equations, least norm solutions. 07 Sept

  4. Orthogonal matrices and their properties, Gram—​Schmidt orthogonalization and the QR factorization. 08 Sept

  5. Function spaces. 15 Sept

Additional resources: Notes on Orthogonality, and related Matlab resources

Reading material: Chapter 3 of Gilbert Strang’s Linear Algebra, except section 3.5 on the fast Fourier transform

Determinants

  1. Basic definition and properties without explicit computations. 21 Sept

Additional resources: Notes on Determinants

Reading material: Sections 4.1 and 4.2 of Gilbert Strang’s Linear Algebra, and the parts of Section 4.4 on pivots and volume

Eigenvalue problems

  1. Definitions and examples, relation between algebraic and geometric multiplicity of an eigenvalue. 26 Sept

  2. Diagonalization of a matrix, powers of a matrix. 28 Sept

  3. Cayley—​Hamilton theorem, computing the Fibonacci sequence via eigenvalue machinery, Hermitian matrices, spectral theorem. 29 Sept

  4. Properties of Hermitian and unitary matrices, the spectral theorem. 05 Oct

  5. Similarity transforms and connections to change of variables. 06 Oct

Additional resources: Notes on Eigenvalue problems, and related Matlab resources

Reading material: Sections 5.1 to 5.3 (up to Markov matrices), Section 5.5, and the first part of Section 5.6 (similar matrices and change of basis; not the Schur and Jordan forms) of Gilbert Strang’s Linear Algebra

Evaluation

  1. Quiz 1 [25] 31 Aug 2026

  2. Quiz 2 [25] 12 Oct 2026

  3. Endsem exam [50] 11 Nov 2026

Outline

Broad course contents

  1. Linear system of equations

  2. Vector spaces

  3. Orthogonality

  4. Determinants

  5. Eigenvalue problems

  6. Positive definite matrices

  7. Singular value decomposition

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

Reference books

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

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