Mathematics for Data Analysis and Modeling

1. Basic Mathematical Concepts and Definitions

Let's start with some basic definitions and concepts we'll use later. Consider the idea of a function, a numerical sequence, and its sum, and also understand what a coordinate system's basis is.

What is a Function?

What are Number Series?

Sum of Elements of The Series

Challenge: Calculating Sum of Geometric Progression

Vectors and Matrices

2. Linear Algebra

The simplest and most commonly used type of relationship is the linear relationship. Linear algebra is a branch of higher mathematics entirely devoted to linear functions and linear spaces. Let's look at some of the most important topics in linear algebra: vectors, matrices, solving linear equations, and solving the spectral problem for matrices.

Numerical Operations on Vectors and Matrices

Challenge: Calculate the Matrix Multiplication Result

Matrix Determinant

Scaling Factor of the Linear Transformation

Challenge: Figures' Linear Transformations

Inversed and Transposed Matrices

System of Linear Equations

Challenge: Solving the Task Using SLE

Eigenvalues and Eigenvectors

3. Mathematical Analysis

Mathematical analysis is a discipline that allows you to analyze functions according to various criteria. Consider how to check numerical sequences for convergence, find the maximum/minimum values of functions, solve nonlinear equations, and use integrals to solve applied problems.

Derivative of the Function

Partial Derivative of the Function

Challenge: Solving Task Using Derivative

Optimization Problem

Challenge: Solving the Optimisation Problem

Gradient Descent Method

Challenge: Optimising Function Of Multiple Variables

Challenge: Solving the Task Using SLE

Task

We have already considered how to solve the SLE using inversed matrix. But we can also use np.linalg.solve(A, y) method that calculates the solution of the SLE:
A * x = y.
Your task is to solve the system using both these methods and compare the results: