CS6015: Linear Algebra and Random Processes

Course information

Instructor
Prashanth Wednesday 2:30pm to 3:30pm BSB 314
TAs
Abhishek Monday 2:30pm to 3:30pm DON lab
Hardi Tuesday 2:30pm to 3:30pm DON lab
Ditty Wednesday 2:30pm to 3:30pm AIDB lab
Jom Thursday 2:30pm to 3:30pm DON lab
Purvi Friday 2:30pm to 3:30pm DON lab

Course Content

Linear Algebra

Matrices

Matrix Multiplication, Transposes, Inverses, Gaussian Elimination, factorization A=LU, rank

Vector spaces

Column and row spaces, Solving Ax=0 and Ax=b, Independence, basis, dimension, linear transformations

Orthogonality

Orthogonal vectors and subspaces, projection and least squares, Gram-Schmidt orthogonalization

Determinants

Determinant formula, cofactors, inverses and volume

Eigenvalues and Eigenvectors

Characteristic polynomial, Diagonalization, Hermitian and Unitary matrices, Spectral theorem, Change of basis

Positive definite matrices and singular value decomposition

Random processes

Preliminaries

Events, probability, conditional probability, independence, product spaces

Random Variables

Distributions, law of averages, discrete and continuous r.v.s, random vectors, Monte Carlo simulation

Discrete Random Variables

Probability mass functions, independence, expectation, conditional expectation, sums of r.v.s

Continuous Random Variables

Probability density functions, independence, expectation, conditional expectation, functions of r.v.s, sum of r.v.s, multivariate normal distribution, sampling from a distribution

Convergence of Random Variables

Modes of convergence, Borel-Cantelli lemmas, laws of large numbers, central limit theorem, tail inequalities

* Advanced topics (if time permits)

Markov chains, minimum mean squared error estimation

Grading

  • Mid-term (Linear Algebra concepts): 30%

  • Final exam (Probability concepts): 30%

  • Quizzes: 20% (Best 5 out of 8)

  • Programming Assignments: 20%

Target Audience

Masters (M.Tech/M.S.) and Ph.D. students

Important Dates

Problem Sets Quizzes Tutorials Mid-Sem End-Sem
Aug 7 Aug 16 Aug 11 When: 10am to 12pm, Sep 24      Where: CS34 and CS36 When: 1pm to 4pm, Nov 15      Where: CS34 and CS36
Aug 18 Aug 28 Sep 1
Aug 25 28 Sep 5 Sep 20
Sep 11 Sep 18 Oct 13
Sep 28 Oct 6 Oct 23
Oct 10 Oct 17 Nov 10
Oct 20 Oct 27
Oct 27 Nov 3

Textbooks

  • Linear algebra and applications by Gilbert Strang

  • Probability and random processes by Geoffrey Grimmett and David Stirzaker

Additional references:

Schedule

Part I: Linear Algebra
Lecture number Topics Covered Section reference
(Strang's book)
Lecture 1 Course Organization
Motivation for studying linear algebra
Lecture 2 Geometry of linear equations - row picture, col picture
Vector space, subspace - definition, examples Linear combinations, linear independence
1.2
Lecture 3 Transpose - properties
Inverse
Gaussian elimination
1.3
Lecture 4 Computational cost of elimination
Matrix multiplication - four view
Gauss-Jordan method
1.4
Lecture 5 Factorization A=LU and A=LDU
Row exchanges, permutation matrices
Uniqueness of LU/LDU factorization for invertible matrices
1.5
Lecture 6 Column and null spaces
Null space computation by solving Ax=0
Pivot and free variables, special solutions
row reduced echelon form
2.1, 2.2
Lecture 7 Dimension = number of vectors in any basis
Rank, row-rank = col-rank
rank + nullity = number of columns
2.3, 2.4
Lecture 8 Orthogonal vectors and subspaces
Row space orthogonal to null space
3.1
Quiz 1
Lecture 9 Projection onto a line
Projection onto a subspace
3.2
Lecture 10 Least squares data fitting
Orthonormal vectors
3.3
Lecture 11 Four fundamental subspaces (again)
Least squares data fitting
2.4, 3.3
Lecture 12 Orthogonal bases
Gram Schmidt algorithm
Factorization A=QR
3.4
Quiz 2
Lecture 13 Linear transformations:
definition, matrix representation
2.6
Lecture 14 Composition of linear transformations
Change of basis
Change of basis: Section 46 of Halmos's text
Lecture 15 Similarity of transformations Section 47 of Halmos's text
Quiz 3
Lecture 16 Determinants: Properties, Formula 4.2, 4.3
Lecture 17 Determinants: Cofactors, Applications in graphs 4.3, 4.4
Graph applications: Section 3.3 of Goodaire's text
Lecture 18 Eigenvalues and Eigenvectors 5.1
Lecture 19 Similarity and diagonalization 5.2
Lecture 20 Spectral theorem for real symmetric matrices:
case when eigenvalues are distinct
5.5
Lecture 21 Complex vector space, Hermitian and unitary matrices 5.5
Quiz 4
Lecture 22 Schur's theorem
Spectral theorem as a corollary
5.5
Lecture 23 Singular value decomposition 6.3
Mid-sem Exam

Part II: Random Processes
Lecture number Topics Covered Section reference
(Grimmett's book)
Lecture 1 Events as sets, probability spaces 1.2
Lecture 2 Cardinality, countability and infinite sums Sections 4 and 5 of Manjunath Krishnapur's notes
Lecture 3 Properties of probability measure 1.3
Lecture 4 Conditional probability 1.4
Quiz 5
Lecture 5 Independence
Gambler's ruin
1.5
Lecture 6 Random variables and distribution function 2.1
Lecture 7 Uncountable probability spaces, distribution functions Section 14 of Manjunath Krishnapur's notes
Lecture 8 Law of averages - Bernstein's inequality 2.2
Lecture 9 Discrete r.v.s - definition and examples
Independence
3.1
Quiz 6
Lecture 10 Discrete r.v.s - expectation, higher moments and examples 3.3, 3.5
Lecture 11 Discrete r.v.s - joint distribution function
Covariance, correlation
3.6
Lecture 12 Discrete r.v.s - conditional distribution and expectation 3.7
Lecture 13 Discrete r.v.s - conditional expectation
Sums of r.v.s
3.7, 3.8
Quiz 7
Lecture 14 Continuous r.v.s - p.d.f., independence, expectation 4.1, 4.2, 4.3
Lecture 15 Continuous r.v.s - examples 4.4
Lecture 16 Continuous r.v.s - dependence 4.5
Quiz 8
Lecture 17 Continuous r.v.s - conditional distributions and expectation 4.6
Lecture 18 Continuous r.v.s - functions of r.v.s 4.7
Lecture 19 Continuous r.v.s - change of variables 4.7
Lecture 20 Continuous r.v.s - multivariate normal distribution 4.9