CS6015: Linear Algebra and Random Processes
Course information
When: JulNov 2019
Lectures: Slot D
Where: CS36
Teaching Assistants: Nirav Bhavsar, Ajay Pandey, Nithia V, Monisha J
Office Hours: Wed 2:30pm to 3:30pm
Course Content: See Schedule below
Grading
Midterm (Linear Algebra concepts): 30%
Final exam (Probability concepts): 30%
Quizzes: 20% (Best 4 out of 6)
Programming Assignments: 20%
Target Audience
Masters (M.Tech/M.S.) and Ph.D. students
Important Dates
Quizzes: Aug 13, Aug 29, Sep 19, Oct 14, Oct 30, Nov 11
Midterm: 10am to 12pm, Sep 28
Endsem: 2pm to 5pm, Nov 20
Lecture Notes
Linear Algebra (incomplete)
Probability
Textbooks
Linear Algebra:
Linear algebra and applications by Gilbert Strang
Linear Algebra: Pure and Applied by Edgar Goodaire
Probability
Probability and random processes by Geoffrey Grimmett and David Stirzaker
ECE 313 UIUC course lecture notes by Bruce Hajek
Additional references:
Introduction to Probability by Bertsekas and Tsitsiklis
Introduction to probability models by Sheldon Ross
Schedule from 2017
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 GaussJordan 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, rowrank = colrank 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 
Midsem 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 