What is linear algebra?

Have you ever had to solve systems of equations with lemons, oranges, and apples from social media? Have you ever calculated the scope of your life, going (hopefully) uphill? Does the word pixels ring a bell to you in terms of camera capacity? Do vectors sound familiar: be it from physics, math, SVG image format, or the villain from Despicable Me? Believe it or not, this is all linear algebra.

The idea behind linear algebra

Do you remember working with simple math class equations like $2x + 3 = 7$? Well, that's where linear algebra begins. Linear algebra is a branch of mathematics that helps you understand and solve these types of equations, but it takes it a step further. It deals with a whole world of equations involving multiple variables, like $2x + 3y - z = 10$ or even whole systems of them, like:

$$\begin{cases} 2x + 3y - 4z + 5w = 7 \\ 1x + 2y + 0z - 3w = 6 \\ 3x - 1y + 2z + 4w = 2 \\ 0x + 5y + 1z - 2w = 8 \\ \end{cases}$$Linear algebra equips you with tools to handle these more complex problems, allowing you to find solutions to systems of equations, and it's not just about numbers. It can help you manipulate and analyze various data, whether it's organizing information, creating 3D computer graphics, or even deciphering patterns in genetic data.

To do this, linear algebra uses simple and complicated structures, methods, and concepts to deal with such problems, which you will learn step by step in the following topics. The main thing is to be eager to learn and concentrate on what you learn.

Definition of linear algebra

As usual, there is not a unique definition of areas in mathematics, and mainly, they include concepts that you should learn after getting familiar with the definition. So, don't be scared if you read unfamiliar terminology in these definitions.

Our beloved Wikipedia defines it this way: Linear algebra is the branch of mathematics concerning linear equations, such as:

$a_1x_1 + \cdots + a_nx_n = b,$

linear maps such as:

$(x_1, \dots, x_n) \mapsto a_1x_1 + \cdots + a_nx_n,$

and their representations in vector spaces and through matrices.

On the other hand, Wolfram MathWorld gives a more concise definition: Linear algebra is the study of linear sets of equations and their transformation properties.

The general idea is that linear algebra is that part of algebra that deals with linear stuff, where linear can be interpreted:

• numerically, as equations involving only the first power,
• structurally, as a vectors or matrices,
• or geometrically as straight lines, planes, or higher-dimensional spaces of this type.

Applications of linear algebra

Linear algebra is a versatile tool that profoundly impacts the real world. Its applications span various fields, each showcasing its power in unique ways.

• In computer graphics, linear algebra plays a pivotal role in creating immersive 3D environments for video games. It leverages matrices and vector transformations to manipulate object positions and orientations, giving rise to visually stunning experiences.
• Data science relies heavily on linear algebra, with linear regression being a prime example. Linear regression models relationships between variables by employing matrix calculations, enabling predictions based on historical data. Clustering algorithms employ linear algebra to group similar data points, revolutionizing customer segmentation and market analysis.

  

• Engineering benefits from linear algebra, too, particularly in electrical circuit analysis. Engineers use linear equations, founded on principles like Kirchhoff's laws, to design and optimize electrical systems efficiently.
• Cryptography, a field of paramount importance in the digital age, relies on linear algebra for secure data transmission. Techniques and algorithms employ mathematical operations on matrices and vectors to protect sensitive information.
• In robotics, linear algebra is integral to solving complex problems like inverse kinematics, enabling robots to accurately calculate the precise joint angles required to position their end effectors. This is vital in applications ranging from manufacturing to surgical robotics.

This field of mathematics continues to shape and enhance our modern world in remarkable ways, so studying it is a must for everyone who strives to become a professional in the IT fields mentioned above.

Conclusion

In conclusion, linear algebra is not just a theoretical concept; it's an indispensable tool with wide-reaching applications in our daily lives. Whether you're exploring 3D virtual worlds in video games, making data-driven decisions in data science, or guiding the movements of robots, linear algebra quietly underpins these advances. Its significance in our ever-evolving world is undeniable, making its study necessary for those venturing into fields like information technology. So, let's start this exploration by uncovering the mysteries of linear algebra and appreciating the mix of formal and informal aspects that make this mathematical world so captivating.