Applications open now for next batch of the Diploma Program | Applications close: 8 July, 2022

Applications open now for next batch of the Diploma Program | Applications close: 8 July, 2022

Mathematics for Data Science II

This course aims to introduce the basic concepts of linear algebra, calculus and optimization with a focus towards the application area of machine learning and data science.

by Sarang S Sane

Course ID: BSCMA1003

Course Credits: 4

Course Type: Foundational

Recommended Pre-requisites: None

What you’ll learn

Manipulating matrices using matrix algebra.
Performing elementary row operations.
Using Gaussian Elimination: Solving systems of linear equations. Find out whether a set of vectors are linearly independent. Writing down a set of dependencies in case vectors are not linearly independent. Finding subspaces along with their bases and ranks.
Finding distances and angles using norms and inner products.
Obtaining orthonormal basis using the Gram-Schmidt process.
Finding maxima and minima of single variable functions using derivatives.
Finding maxima and minima of multivariate functions using vector calculus.

Course structure & Assessments

12 weeks of coursework, weekly online assignments, 2 in-person invigilated quizzes, 1 in-person invigilated end term exam. For details of standard course structure and assessments, visit Academics page.

WEEK 1 Vector and matrices - Vectors; Matrices; Systems of Linear Equations; Determinants (part 1); Determinants (part 2)
WEEK 2 Solving linear equations - Determinants (part 3); Cramer's Rule; "Solutions to a system of linear equations with an invertible coefficient matrix"; The echelon form; Row reduction; The Gaussian elimination method
WEEK 3 Introduction to vector spaces - Introduction to vector spaces; Some properties of vector spaces; Linear dependence; Linear independence - Part ; Linear independence - Part 2
WEEK 4 Basis and dimenstion - What is a basis for a vector space?; Finding bases for vector spaces; What is the rank/dimension for a vector space; Rank and dimension using Gaussian elimination
WEEK 5 Rank and Nullity of a matrix; Introduction to Linear transformation - The null space of a matrix : finding nullity and a basis - Part 1; The null space of a matrix : finding nullity and a basis - Part 2; What is a linear mapping - Part 1; What is a linear mapping - Part 2; What is a linear transformation
WEEK 6 Linear transformation, Kernel and Images - Linear transformations, ordered bases and matrices; Image and kernel of linear transformations; Examples of finding bases for the kernel and image of a linear transformation
WEEK 7 Equivalent and Similar matrices; Introduction to inner products - Equivalence and similarity of matrices; Affine subspaces and affine mappings; Lengths and angles; Inner products and norms on a vector space
WEEK 8 Orthogonality, Orthonormality; Gram-schmidt method - Orthgonality and linear independence; What is an orthonormal basis? Projections using inner products; The Gram-Schmidt process; Orthogonal transformations and rotations
WEEK 9 Multivariavle functions, Partial derivatives, Limit, continuity and directional derivatives - Multivariable functions : visualization; Partial derivatives; Directional derivatives; Limits for scalar-valued multivariable functions; Continuity for multivariable functions; Directional derivatives in terms of the gradient
WEEK 10 Directional ascent and descent, Tangent (hyper) plane, Critical points - The directional of steepest ascent/descent; Tangents for scalar-valued multivariable functions; Finding the tangent hyper(plane); Critical points for multivariable functions
WEEK 11 Higher order partial derivatives, Hessian Matrix and local extrema, Differentiability - Higher order partial derivatives and the Hessian matrix; The Hessian matrix and local extrema for f(x,y); The Hessian matrix and local extrema for f(x,y,z); Differentiability for Multivariable Functions; Review of Maths - 2
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About the Instructors

Sarang S Sane
Assistant Professor, Department of Mathematics, IIT Madras

I completed my B.Stat. (Hons.) and M.Stat. from the Indian Statistical Institute, Kolkata in 2004 and my Ph.D. from TIFR, Mumbai in 2010. I was a postdoctoral fellow in TIFR, a visiting assistant professor in the University of Kansas and very briefly an INSPIRE faculty fellow in IISc, Bengaluru before I joined the mathematics department in IITM in 2015.