Course

Module-1

Linear Algebra : Linear dependence/independence of vectors, Rank of matrices, System of linear equations, Solution of system of linear equations, Vector spaces, Subspaces, Generating set, Basis, Dimension of vector spaces, Linear mapping, Transformation matrix, Basis change, Image and Kernel of linear mapping, Affine spaces, Norm of a vector space, Dot product, Quadratic form, symmetric positive definite matrices, Length, angle and orthogonality, Orthonormal basis, Inner product of functions, Orthogonal projections, Gram Schmidt orthogonalization, Eigen values andvectors,Matrixdecomposition:Choleskydecomposition,Eigendecomposition,Singularvaluedecomposition,Matrixapproximation.

Module-2

Vector calculus : Differentiation and Taylor’s series expansion of uni variate functions, Partial differentiation, chain rule, Gradient of vector function(Jacobian), Gradient of a vectors with respect to a matrix, Identities for computing gradients, Back propagation and automatic differentiation, Gradients in deep neural networks, Higher order partial derivatives, Hessian, Taylor’s series expansion of multivariate functions, Vector calculus for physical field problems, divergence and curl of vector fields, rotational and irrotational vector fields, Conservative vector fields, Vector integral calculus, line, surface and volume integrals, Stoke’s theorem, Green’s theorem and Gauss divergence theorem,
Probability: Introduction to Probability concepts, one dimensional and two dimensional Random variables, Jointly Distributed Random Variables, Conditional Distributions.

Module-3

Solution using iterative methods : Gauss Seidel, SOR(pointandline), Conjugate gradient, BiCG Stab, GMRES, Solution of systems of nonlinear algebraic equations.
Interpolations : Newton, Stirling, Lagrange, Richardson, Quadratic and Cubic splines,Inverse interpolation.
Numerical Differentiation ; Numerical integration:Higher-OrderNewton-Cotes formulas, Romberg integration,multiple integrals.

Lab Session

• Verify vector dependence/independence, find rank, basis, dimension, eigenvalues/eigenvectors, and perform matrix decompositions (Cholesky, Eigen, SVD) using MATLAB / Python with NumPy and SciPy.
• Calculate gradients, Jacobians, Hessians, Taylor series expansions, and implement backpropagation and automatic differentiation using MATLAB / PyTorch or TensorFlow in Python.
• Solve linear and nonlinear equations using iterative methods like Gauss-Seidel, SOR, Conjugate Gradient, BiCGStab, and GMRES with MATLAB / SciPy in Python.
• Apply interpolation (Newton, Lagrange, Stirling, splines), numerical differentiation, and integration (Newton-Cotes, Romberg, multiple integrals) using MATLAB / SciPy in Python.

Course Outcomes

• CO1: Capability to understand mathematical notations and solve problems in linear algebra.
• CO2: Capability to do gradient evaluations of objective functions in optimization problems
• CO3:Capabilityto do scalarand vector field operations
• CO4:Capability for solving mathematical equations model ledas differential equations
• CO5:Solve problems indistribution of random variables and parameter estimation.

• Advanced Engineering Mathematics–ErvinKreyszig,10^thedition(2015),Wiely.
• Linear algebra and its application –Gilbert Strang,5^thedition(2018),Cengage Learning.
• Miller and Freund’s-Probability and Statistics for Engineers–Richard A Johnson,7^thedition(2008),Pearson.
• Differential equations with applications and Historical notes-George F Simmons-Tata Mc GrawHill,3^rdedition (2016),Taylor& Francis.
• Mathematics for Machine Learning–Marc Peter Deisenroth,A.Also Faisal,Cheng SoonOng,Cambridge UniversityPress (2020)
• Machine leaning:A probabilistic perspective,KevinMurphy,MITPress(2012).