Interactive Online textbook on Linear Algebra

The Delft Institute of Applied Mathematics (DIAM), in close collaboration with the Programme of Innovation in Mathematics Education (PRIME), has launched a new interactive online textbook on linear algebra, designed specifically for engineering students at the bachelor level.

According to the institute communication release, "The book introduces the fundamentals of linear algebra from a geometrical perspective, starting with vectors, lines, and planes, before moving on to core topics such as systems of linear equations, matrix arithmetic, eigenvalues and eigenvectors, and orthogonality. More advanced subjects, including symmetric matrices and discrete dynamical systems, are explored in later chapters.

Students can explore concepts hands-on through applets, while a wide range of embedded exercises with personalised feedback ensures active engagement and deeper understanding. The book has already been integrated into several interfaculty linear algebra courses at TU Delft."

Here is the link to the book:

https://interactivetextbooks.tudelft.nl/linear-algebra/index.html

Here is the repository of all the interactive open books from TU Delft.

https://books.open.tudelft.nl/home/catalog/category/interactive-textbooks

hashtag#mathematics #linearalgebra #TUDelft #MathematicsEducation #AppliedMathematics Delft University of Technology TU Delft | PRIME

Mathematical Reliability in Aviation Safety

✳️ The "Course Gallery" is a hands-on project exhibition at our college, where various departments showcase thematic display items created by students. This year, we had this during the first week of October. ✳️

🛩 Our team thematically explained the statistical reliability testing process by exhibiting a model Airplane Disaster safety system. 🛩

✈️ The tragic incident of the recent Ahmedabad plane crash prompted us to reflect on the importance of aviation safety. At that time, numerous discussions about various passenger protection innovations emerged in the media in the context of air crashes. What is crucial in deciding about a particular safety system is the statistically significant success rates achieved in multiple iterations during the testing phase, where mathematics plays a critical role.🚒

💡 We came across an innovative idea proposed by Ukrainian engineer Vladimir Tatarenko in 2016. His idea is as follows: Under normal conditions, the cabin remains attached to the plane. When an emergency occurs, the emergency switch is pressed. The motor activates and releases the cabin lock. Simultaneously, the parachute opens, and the cabin lands slowly and safely on the ground.🪂

📚 Our team believed that a practical demonstration model on this concept would be an effective way to explain the role of statistics in such critical areas. The details are provided in the image below. 🛠

The inventor conducted a survey, which showed that people are ready to pay extra for this safety idea. 

☢️ But there are challenges too: (1) The detachment mechanism could compromise the aircraft's structural integrity. (2) The falling cabin could pose a danger to people and property on the ground, and its landing could be affected.🚫

However, this innovation can serve as a stepping stone for creating better solutions to prevent loss of life during plane crashes. 

#mathematics #statistics #riskassessment #airplanesafety #aviationsafety #appliedmathematics #mathprojects

Application of Linear Algebra in Industrial Automation

Abstract: Linear algebra is a major topic in mathematics that deals with vectors, matrices, and linear functions, among other concepts. Industrial automation refers to the use of machines, robotics, and control systems to perform tasks that would otherwise be done by humans, thereby improving quality, productivity, and safety in industrial applications and developments. This article explores the significant reliance of industrial automation on linear algebra as its foundational basis. This article primarily focuses on explaining two crucial topics: robotics and artificial intelligence, as well as the application of linear algebra in these fields.

Introduction

Linear Algebra is an inevitable part of mathematics that deals with matrices, vectors, tensors, vector spaces, linear functions, and mappings, to name a few. This works like algebra, but instead of numbers, vectors are used. This is one of the important topics that forms the foundation of many other fields, including physics, chemistry, industrial automation, psychology, and engineering. Linear Algebra is used in many different fields due to its versatile concepts, such as matrices and vectors, which aid in simplifying and understanding complex problems.

The areas in Industrial Automation where linear algebra is applied are, 

• Robotics

• Artificial Intelligence

Robotics

Robotics is a field of engineering and computer science that deals with the design, manufacture, and operation of robots by applying concepts from computer science, mechanical engineering, and other related domains. Many industries, such as the automobile industry, use small robots to perform repetitive tasks. This field aims to create machines that can assist humans in various tasks. 

Vectors are quantities that contain both magnitude and direction. They are used to represent the movements made by the robots in 2D and 3D space. 

Few Applications of Linear Algebra in Robotics

The inverse of a matrix is used to solve a system of linear equations. This concept is applied in robotics to perform fundamental tasks, such as control, motion planning, and kinematics.

Singular Value Decomposition is an exciting concept in linear algebra. It is a technique used to break down complex matrices into smaller component matrices. In Robotics, this technique is used in motion planning to find the best configuration for the robot. This technique is used to evaluate the feasible trajectories by analyzing the workspace of the robot. SVD is also used in the analysis of applications in robotics.

Transformation is another exciting topic. It is a function from one vector space to another that does not alter the linear structure of either vector space. Matrices are used in transformations. This enables robots or machines to calculate their position and altitude in a 3D vector space.

Artificial Intelligence

Artificial Intelligence is a type of concept in Industrial automation that requires human intelligence. These tasks include reasoning, problem-solving, and language understanding. Artificial intelligence enables machines to analyze data, make informed decisions, and enhance their performance.

Application of Linear Algebra in Artificial Intelligence

Vectors are used to represent data points and attributes with magnitude and direction. It is used as a fundamental tool for capturing and manipulating information about data points, enabling the representation of numerical features in machine learning models.

Matrices are rectangular arrays of numbers that form a relation between vectors. Matrices are used to represent complex structures and capture intricate relationships within data sets.

In data representation, linear algebra serves as a tool for representing and manipulating data. Linear algebra enables the representation of text data as vectors and image data as matrices.

The data representations permit AI algorithms to process information effectively and make sense of textual and visual data. Linear algebra facilitates the handling of diverse and unstructured data types, including text, speech, and image data.

Optimization utilizes linear algebraic tools to solve optimization problems through systems of linear equations. Many AI algorithms rely on this concept to find solutions. Least squares regression and gradient descent are techniques that heavily depend upon linear algebra. Recent advancements in optimization have focused on addressing challenges like non-convexity and large-scale optimization problems. Stochastic gradient descent with acceleration in an algorithm that has been developed to train deep neural networks more efficiently. Concepts of optimization in linear algebra, such as interior point methods and conjugate gradient methods, have been extended to handle complex AI models with millions of parameters.

Neural Networks are a vast topic in AI and machine learning. It can be said to be the heart of AI. Linear Algebra serves as the foundation for neural networks. Neural networks have many layers, which can be represented in a matrix transformation. Matrix multiplication, elementary row operations, and activation functions are utilized in the forward and backward propagation steps during the training of neural networks. Image classification, natural language processing, and speech recognition are a few areas in which neural networks have proven highly successful.

Conclusion

From designing efficient robotic movements and improving image processing in quality control to optimizing resource allocation and enhancing machine learning models, linear algebra enables automation systems to function with precision and efficiency. As industries continue to embrace automation and AI-driven solutions, the application of linear algebra will remain indispensable in driving innovation, improving productivity, and ensuring seamless operations in modern manufacturing and industrial processes.

References

https://www.techtarget.com/whatis/defenition/robotics

https://www.cs.miami.edu/home/visser/csc398-files/linearalgebra-notes.pdf

https://www.linkedin.com/pulse/linear-algebra-fuels-artificial-intelligence-kayode-odeyemi

https://www.mirlabs.org/isda23/ISDA23_SS14.pdf

https://www.kiet.edu/blog/computer-science-information-technology/transforming-ai-and-%20machine-learning-the-crucial-role-of-linear-algebra/

https://medium.com/@rajat01221/mathematics-of-neural-networks-from-linear-algebra-to-%20backpropagation7bb72b73f49c

https://studywolf.wordpress.com/2017/06/09/deriving-a-robots-transform-matrices/

https://turcomat.org/index.php/turkbilmat/article/download/11829/8642/20975#:~:text=Algorithm%20of%20New%20Conjugate%20Gradient,0%2C1%2C2%2C%E2%80%A6