Contextualization

Theoretical Introduction

Matrices are fundamental and powerful concepts in mathematics. Originating from theories of systems of linear equations, they are much more than simple rectangular arrangements of numbers. Through matrices, we can represent linear transformations, solve systems of equations, and deal with multidimensional quantities efficiently.

Operations with matrices, such as addition, subtraction, and multiplication, are as valid and indispensable as the operations we perform with real numbers. They allow for data set manipulations, system changes, and the creation of new sets from existing matrices.

Multiplying matrices, for example, is not as simple as multiplying numbers. In matrix multiplication, we consider each row of one matrix and each column of another, resulting in a third matrix. This process is essential in various fields of knowledge, such as computer graphics, where we use it to perform transformations like rotation, scaling, and translation.

Importance and Applications of Matrices

Matrices are extremely important tools in computer science, physics, economics, engineering, and many other areas of knowledge. They are used in computer graphics to process linear transformations, in physics to solve systems of linear equations, and in economics to model and solve production and consumption problems.

In the digital world, matrices are used in various functionalities. The pixels on your televisions and monitors are controlled by matrices. When you play a video game and the character moves or rotates, matrices are behind those movements.

Finally, matrices are used in cryptography, making our data secure on the internet. Whether making an online purchase or sending a private message on social media, matrices help ensure that your information reaches its destination without being intercepted.

Resources for Further Study:

• Book: "Linear Algebra with Applications", Anton Howard (This book provides a clear and concise approach to linear algebra and its applications, making it an excellent resource for delving deeper into the study of matrices.)

• Website: Khan Academy, Linear Algebra Course (Khan Academy is a free, high-quality platform for self-directed learning. Its Linear Algebra course is well-explained and covers key concepts of matrix operations.)

• Video: "What are matrices and what are they used for?" from the Matemática Rio channel with Prof. Rafael Procopio (This video provides an excellent introduction to matrices and their applications.)

Practical Activity - "Matrices in Cinema: Creating the Magic of Visual Effects"

Project Objective

This project aims to apply the concepts of matrices and their operations, especially matrix multiplication, in a practical, contextualized, and fun situation: creating a small visual effect for a movie scene! Additionally, the project also seeks to develop socio-emotional skills in students through group work, time management, and problem-solving.

Detailed Project Description

Groups of 3 to 5 students should choose a scene from a favorite movie and create a simple "special effect" through matrix manipulation. This effect can be a rotation, translation, scaling, or any other linear transformation applied to the image.

Required Materials

• Computers or laptops with internet access;
• Image editing software (such as GIMP, Photoshop, or even Paint);
• Software for matrix manipulation (such as the free software GeoGebra, or the Python programming language with the NumPy library);
• An image of the movie scene chosen by the students.

Detailed Step-by-Step for Activity Completion

1. Image Selection: Each group should choose a scene from a movie that all members enjoy.
2. Image Manipulation: The chosen image should be manipulated in an image editing software so that it can be worked on.
3. Image Representation as a Matrix: Each group should then represent the chosen image as a matrix. Each pixel of the image will be a cell in the matrix, and its intensity will be the cell's value.
4. Transformation Creation: Students should create a matrix that represents the transformation they want to apply to the image.
5. Transformation Application: Students should multiply the image matrix by the transformation matrix to obtain the transformed image.
6. Presentation of Results: Finally, students should present the original image and the transformed image, and explain in detail the transformation they performed.

Project Deliverables and Document Writing

At the end of the project, groups should deliver:

1. The original movie scene and the scene with the applied effect.
2. A detailed written report containing:
   • Introduction: Contextualization of the theme, explanation of the relevance of matrices, and description of the project's objective.
   • Development: Detailed explanation of the theory involved with matrices and their operations, detailed description of the step-by-step process for the practical activity, the methodology used for the activity, and the presentation and discussion of the results obtained.
   • Conclusion: Recap of the main points, explicit mention of the learnings obtained, and conclusions drawn from the project.
   • Bibliography: Indication of all sources that aided in the development of the work.

Students should pay close attention to writing the report. This document is as important as carrying out the practical activity, as it is their opportunity to demonstrate understanding, mastery of the subject, and the ability to express themselves clearly and concisely.