Unboxing Linear Algebra — Chapter 1: The Introduction

Linear algebra is a fundamental branch of mathematics that deals with vector spaces, matrices, and their operations. Its applications are vast and can be found in various fields such as physics, computer science, economics, and engineering. In this article, we will explore the importance of linear algebra and introduce the basic concepts and terminology involved.

1. Overview and the Applications

Linear algebra provides a powerful framework for understanding and solving problems that involve linear relationships. It allows us to analyze and manipulate objects such as vectors and matrices, which are fundamental building blocks in many mathematical models and real-world scenarios. Some of the key applications of linear algebra include:

• Computer graphics: Linear algebra is essential for transforming and manipulating images, rendering 3D graphics, and simulating realistic animations.
• Machine learning: Linear algebra plays a vital role in machine learning algorithms, where data is often represented as vectors or matrices, and operations like matrix multiplication and eigendecomposition are employed.
• Physics: Linear algebra is used extensively in physics to describe and solve problems involving physical quantities such as forces, velocities, and electric fields.
• Economics: Linear algebra is applied in economic models to analyze systems of linear equations representing supply and demand relationships, production processes, and market equilibrium.
• Engineering: Linear algebra is utilized in various engineering disciplines, including control systems, signal processing, and electrical circuit analysis.

2. Basic Concepts and Terminology

To delve into linear algebra, it is crucial to understand some basic concepts and terminology. Let’s introduce a few key terms:

Vectors

A vector is a quantity that has both magnitude and direction. It is commonly represented as a column or row of numbers enclosed in square brackets. For instance, a 3-dimensional vector can be represented as:

Matrices

A matrix is a rectangular array of numbers or symbols arranged in rows and columns. It can be thought of as a collection of vectors. Matrices are denoted by uppercase letters. For example, a matrix A can be represented as:

Scalars

Scalars are single numbers and do not have any direction. They can be real numbers or elements from other number systems.

3. Notation and Representation of Vectors and Matrices

In linear algebra, various notations and representations are used to describe vectors and matrices. Here are a few commonly employed notations:

• Boldface lowercase letters (e.g., v) are used to represent vectors.
• Boldface uppercase letters (e.g., A) denote matrices.
• Elements of vectors or matrices are represented by subscripts. For example in the below image, i​ represents the i-th element of vector v, and ij​ represents the element at the i-th row and j-th column of matrix A.

These notations and representations provide a concise and standardized way of working with vectors and matrices, making it easier to express mathematical relationships and perform operations on them.

In the next article, we will delve deeper into vectors and their operations, which will lay the foundation for further understanding of linear algebra concepts. Stay tuned for the next article in this series!

If you found this course helpful and would like to explore more free courses, I invite you to follow my account on Medium and connect with me on LinkedIn. I regularly share valuable content on these platforms.