Linear Algebra - Linear Systems - Part 1

In my previous article, I wrote about the importance of mathematics in the field of artificial intelligence and its subsets: Machine Learning and Deep Learning.

Developing high-impact products in this world requires more than just using ready-made tools. It demands a deep understanding of mathematical foundations. There is a clear difference between using AI services to build solutions and creating AI systems from scratch. The latter path requires sharper technical and mathematical skills.

With that in mind, I decided to talk about the mathematics involved in the world of artificial intelligence, a subject I both admire and study enthusiastically.

The main mathematical areas involved in this field, as mentioned in the previous article, are linear algebra, calculus, and statistics. Each one plays a fundamental role in the structure and functioning of AI models.

In this post, the focus will be on linear algebra, one of the main pillars of modern AI. It is responsible for handling large volumes of data through structures such as matrices, vectors, and tensors, which are essential for representing and manipulating information in the form of images, text, audio, and other formats.

Outside of artificial intelligence, linear algebra also has applications in fields such as computer graphics, engineering, physics, economics, and finance. It is also the foundation of analytic geometry, enabling us to describe and analyze objects and shapes in space using equations and coordinates.

But after all, what is Linear Algebra?

To understand how these concepts operate in practice, it is first essential to understand what linear algebra is.

Linear algebra is a branch of mathematics that studies vector spaces, operations with vectors and matrices, linear transformations, and systems of linear equations. It allows us to understand phenomena in multiple dimensions and to perform mathematical operations in these spaces efficiently and in a structured way.

And what is a linear system?

A linear system is a set of linear equations that share one or more unknowns in common. The definition of a linear system arises from the fact that the equations follow a linear format, that is, the variables are not raised to any power nor multiplied by each other.

Example of a linear system:

5x + y = 3
2x - y = 4

Solving this linear system means finding the values of x and y that satisfy both equations simultaneously.

A linear system can be represented in a two-dimensional or three-dimensional form. The two-dimensional case is represented by a set of two linear equations with two unknown variables:

ax + by = e
cx + dy = f

Where x and y are variables, and a, b, c, d, e, and f are constants. There are several methods used to solve two-dimensional linear systems, such as the substitution method, addition, subtraction, and the equality method. In the following article, I will show how to solve two-dimensional linear systems with real mathematical examples.