# CS6015: Linear Algebra and Random Processes

## Course information

• When: Jul-Nov 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

• Mid-term (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

• Mid-term: 10am to 12pm, Sep 28

• End-sem: 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

• Additional references:

• Introduction to Probability by Bertsekas and Tsitsiklis

• Introduction to probability models by Sheldon Ross

## Schedule from 2017

Lecture number Topics Covered Section reference (Strang's book) Part I: Linear Algebra 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

Lecture number Topics Covered Section reference (Grimmett's book) Part II: Random Processes 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