Mathematical Foundations for Data Science - Science Assignment Help

Assignment Task

Title - Mathematical Foundations for Data Science

Course Description
Vector and matrix algebra, systems of linear algebraic equations and their solutions; eigenvalues, eigenvectors and diagonalization of matrices; Calculus and optimization; Counting principles and combinatorics

List of Topic Title
1 Matrices, rank, determinants, solution of linear systems – analytical techniques
Solution of linear systems (A (m x n) x (n x 1) = b (m x 1); A has rank r.) – just a recapitulation
Solution using Gauss elimination with and without pivoting and operations count
LU decomposition methods

2 Numerical solution for linear systems
LU decomposition methods (Continued)
Iterative methods for linear systems
 

3 Vector spaces and subspaces, basis and dimensions, Linear transformations and properties 
Vector spaces, inner product spaces, properties, LI and LD, bases and dimensions, LT
LT and Rank-Nullity theorem
NS(A), RS(A) and CS(A) – illustration
 

4 Eigenvalues and eigenvectors
Eigenvalues and eigenvectors of special matrices and their properties
Eigenbases and diagonalization
Gerschgorin’s Theorem
Power Method T1: Sec 8.1,8.7, 20.7, 20.8

5-6 Decomposition methods (Eigenvalue, decomposition, QR and SVD)
Gram-Schmidt Orthogonalization procedure
QR decomposition
SVD
Dimensionality reduction
 

7 -8 Application of linear algebra in optimization. Modelling linear programming problem and the basics of Simplex algorithm and sensitivity analysis.
Model a LPP in construction of buildings.
Model the currency arbitrage optimization problem.
Work out the graphical method of solution in the case of 2 variable case
Simplex method for simple cases
Outline how Gauss Jordan produces the inverse matrix.
 

9 Properties of functions
Continuous functions on closed intervals, differentiation (1d case)
Taylor series expansion
Maxima and minima
Integral properties (cdf and pdf, even and off integrands, integration by parts and so on) for 1d
 

10 Calculus of several variables
Review limits, continuity and differentiability (graphically and algebraically)
Vector calculus and some of the identities
Maxima and minima (unconstrained)
Steepest gradient method
Lagrange multipliers (for more number of constraints)
 

11-12 Induction principle
Recursive definition and structural induction
Mathematical induction
Strong induction

13-14 Counting Principles
Basics of counting
Pigeonhole principle
Permutations and combinations
Binomial coefficients and identities

15-16 Advanced counting
Application of recurrence relations
Solving linear recurrence relations

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