Contextualization

Linear systems and their matrix representation are an important pillar in the study of mathematics, in particular in the field of linear algebra. Such concepts are of great value in solving problems involving multiple variables, such as those found in physics, economics, computer science or any other discipline that involves the use of multiple variables.

In this sense, linear systems are sets of equations that, together, relate two or more variables. Matrices, on the other hand, are a powerful mathematical tool for dealing with several numbers at the same time, allowing for the addition, subtraction, and multiplication of sets of numbers in a single operation. Thus, a linear system can be represented by a matrix, simplifying its solution.

Writing linear systems in matrix form, or the process of "matrixizing" a linear system, is an essential skill for efficient manipulation of these systems, allowing the application of linear algebra techniques in solving related problems. This process involves converting linear equations to a matrix representation, which reveals important properties of the system and facilitates its solution.

The importance of these concepts is not limited to the field of mathematics. In the real world, linear systems and matrices are used to model and solve a wide range of problems. In physics, for example, they are used to describe particle systems and equations of state. In economics, they assist in the study of markets and in forecasting models. In computer science, they are used in image processing algorithms and neural networks.

We recommend the following resources for study and further study on this topic:

1. Book: "Fundamentals of Elementary Mathematics: Volume 5 - Linear Algebra", by Gelson Iezzi. This book offers a detailed and in-depth study on the subject, with clear language and practical examples.
2. Videos: "Matemática Rio" channel on YouTube. This channel has a series of videos on the subject, from the introduction to basic concepts to the solution of complex problems.
3. Website: OBMEP Mathematics Portal. Provides a vast amount of study material, including theory, examples, and solved exercises.

Practical Activity: The Enchanted World of Matrices

Project Objective

The objective of this project is to apply the knowledge of linear systems and their matrix representation to solve a real-world problem. Through this work, you will be encouraged to transform linear systems into matrices and see the practical utility of linear algebra.

Project Description

In this project, groups will be challenged to model and solve a real-world problem involving the use of linear systems and matrices. The activity will consist of three main parts:

1. Choosing a Real-World Problem: Groups will identify a real-world problem that can be modeled by a linear system. This can be something in a variety of fields, such as economics, physics, engineering, computer science, etc.

2. Matrixing the Linear System: Using the chosen problem, students will formulate a linear system that represents it. They will then translate this linear system into matrix form.

3. Solving and Interpreting: Finally, the groups will solve the matrix and interpret the solution in the context of the real-world problem they chose.

Required Materials

• Research materials: books, scientific articles, internet, etc.
• Software for mathematical calculations: Excel, Google Sheets or a computer algebra software such as Wolfram Mathematica.

Step-by-Step Activity Guide

1. Group Formation: Groups should consist of 3 to 5 students.

2. Problem Identification: Each group should choose a real-world problem that can be modeled as a linear system. Use books, the internet, and consultations with experts to identify a suitable problem.

3. Formulating the Linear System: Once the problem is chosen, the students will formulate a linear system that represents the problem.

4. Matrix of the Linear System: The next step is to translate the linear system into matrix form. This is the crucial step where the group needs to make sure that the linear system is represented correctly as a matrix.

5. Solving the Matrix: Using a computer algebra software (or even Excel or Google Sheets), the groups solve the matrix.

6. Interpretation of the Solution: After solving the matrix, it is time to interpret what the results mean in the context of the real-world problem they chose.

7. Preparing the Final Report: Each group will prepare a final report describing the chosen problem, the matrix representing the problem, the solution of the matrix, and the interpretation of the result. This report should follow the proposed structure: Introduction, Development, Conclusions and Bibliography used.

   • The Introduction should contextualize the chosen problem and explain why it is relevant. It should also briefly explain what a linear system is, a matrix, and how they relate.

   • In the Development the students should describe the problem in more detail, present the linear system they formulated from the problem and the matrix that represents the system. The solution to the matrix should also be presented in this section. The resolution process, the difficulties encountered and the decisions made during the teamwork should also be reported.

• In the Conclusion the work done and the knowledge acquired must be summarized. Students should also connect the solution found with the real-world problem initially proposed, interpreting what the solutions found mean in the context of the problem. Finally, students should reflect on the process and skills acquired, such as teamwork, time management, problem solving, etc.

• The Bibliography should list all sources that were consulted during the project.

The deadline for submission of the project is one month from the start date. Remember, this activity requires research, reflection, and teamwork. Manage your time wisely and remember that the main goal is to learn and develop as a student and future professional.