Linear algebra is a branch of mathematics that studies vectors, matrices, and linear operators. A vector is a directed length and they are said to be mutually independent if it cannot be represented as a linear combination of the remaining vectors or relations. Orientation is determined for vectors of the same direction – collinear vectors. It is said that vectors are collinear if they can be expressed as a scalar product of each other. Two collinear vectors can be of the same or opposite orientation.

Matrices are any rectangular table of real or complex numbers. In the software solution, the matrices are declared as a structure containing two integer variables, which determine the number of columns and rows of the matrices, two pointers to the structure of the matrices so that a linked list can be made for storage matrix and double-pointer so I can allocate a 2D array containing matrices values.

Matrices and vectors are the basis of programming but are also used in other fields, such as economics, engineering, and physics. Take for example the economics and application of the matrix calculus in solving a system of linear algebraic equations – a linear model of market equilibrium, a model of national income, and a cross-sectoral model (input-output analysis).

## Programming paradigms

The term “programming paradigm” represents the way in which a programming language or programmer designs, ie. designs its programs. There are a number of programming paradigms where there are plenty of overlaps, and programming languages often support more than one so that the issue of “selection” programming paradigms often boil down to taking over certain principles and guidelines that correspond both to the goal that the programmer wants to achieve and to the properties of the selected program language.

Without knowledge of vectors and matrices, a programmer is powerless to do anything. There are various tools on the internet that can be used to calculate matrices, and some of these are Matrix Reshis, OnlineMSchool, WolframAlpha, Symbolab etc. In this whole, it will be presented concretely how to use the Matrix Reshis web application. Matrices are used in everyday life more than people could imagine. When matrices are applied in real life it can save money and find a solution easier. Matrices can be applied in programming, physics, robotics, statistics, video games, etc.

Systems of linear equations are often encountered in everyday life. One useful application of the same is used in chemistry in the equation of chemical equations. Namely, when a chemical reaction occurs, the molecules we call reactants to combine to form a new molecule we call products. Here the notion of a balanced chemical equation is introduced – an algebraic equation that gives the relative number of reactants and products in the reaction and has the same number of atoms of each type on both the left and right sides. The technique of determining the number of reactants and products is carried out by solving homogeneous systems of linear equations.

Computer graphics is a computer field that includes the creation, storage, and processing of image content via a computer. It began to thrive in the 1960s the first interactive creation of graphic content was enabled and it has been developing very fast since then. The computer has a wide application in science, engineering, art, and especially in the field of entertainment: movies and video games. It is represented in the film industry to create various effects, animation and other manipulations over moving images. When creating complex scenes, We are often in a situation where we want to move an object, resize it, or rotate it around a point. Using the matrices calculus and coordinates, each of the mentioned the operation can be displayed as a single matrix. Performing such an operation is reduced to matrix multiplication. Namely, if we want to move a point, we simply add a translation matrix to it, multiply the point we want to rotate by the rotation matrices, and so on. For this reason, it is a matrice the display of these operations is extremely convenient in graphics applications and is so common that forms the basis of performing graphics operations and the core of all popular libraries for working with graphics.

### Final thoughts

In geometry, physics, economics, and various technical disciplines it is necessary to a scalar, ie. real and complex numbers, introduce vectors as well. We describe forces by quantities that have a direction in space and a number that gives strength, which are the properties of vectors in three-dimensional geometric space. The simplest functions that work on vectors are linear. Unlike functions that associate numbers with numbers, functions that associate vectors with vectors are called operators. Linear operators on vector spaces are given by number schemes – these are matrices. Matrices are calculated so that they follow linear operators. Like two functions can be added, so two linear operators can be added. Operation among the matrices that correspond to the summation of the operators is the summation of the matrices. The summation of matrices is defined as follows (A + B) ij = (A) ij + (B) ij. For the sum of matrices to be defined, matrices A and B must be of the same type. The result is again a matrix of the same type (m, n). Similarly, a linear operator can be multiplied by a scalar, and the multiplication of matrices by a scalar corresponds to this.

From this text, we can conclude that the use of matrices and vectors is very wide and that many use them in the performance of daily duties.

It is important to emphasize that many of these tutorials you can find on the internet belong to a series of tutorials published on Statlect, which you can check if you click on statlect.com.