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

Inversed and Transposed Matrices

Matrix Transposition

In linear algebra, the transpose of a matrix is an operation that flips the matrix over its diagonal, resulting in a new matrix where the rows become columns, and the columns become rows. The transpose of a matrix A is denoted as A^T.
We can provide transponation of matrix in Python in two different ways:

Using .T attribute of np.array class;
Using np.transpose() method.

Matrix Inversion

In linear algebra, the inverse of a square matrix A is a matrix denoted as A^-1, which, when multiplied by the original matrix A, results in the identity matrix I. The inverse of a matrix exists only if the determinant of the matrix is non-zero.

Note
An identity matrix, denoted as I, is a square matrix with ones on the main diagonal (from the top left to the bottom right) and zeros elsewhere. In other words, all elements in the main diagonal are equal to 1, while all other elements are equal to 0.

It's important to admit that A * A^-1 = A^-1 * A = I - the multiplication operation by the inversed matrix is commutative.

In Python, you can use the .inv() method of the np.linalg module to calculate the inverse of a matrix. Here's an example:

¿Todo estuvo claro?

Sección 2. Capítulo 6

Contenido del Curso

Mathematics for Data Analysis and Modeling

Inversed and Transposed Matrices

Matrix Transposition

Using .T attribute of np.array class;
Using np.transpose() method.

Matrix Inversion

Note
An identity matrix, denoted as I, is a square matrix with ones on the main diagonal (from the top left to the bottom right) and zeros elsewhere. In other words, all elements in the main diagonal are equal to 1, while all other elements are equal to 0.

It's important to admit that A * A^-1 = A^-1 * A = I - the multiplication operation by the inversed matrix is commutative.

In Python, you can use the .inv() method of the np.linalg module to calculate the inverse of a matrix. Here's an example:

¿Todo estuvo claro?

Sección 2. Capítulo 6