This book provides a comprehensive exploration of the fundamental principles of linear algebra, with a particular focus on different types of systems and matrices, including: Indeterminate systems; Determined systems; Overdetermined systems; Underdetermined systems; Eigenvalue Decomposition (EVD); Singular Value Decomposition (SVD); and Moore-Penrose pseudo-inverse. Each topic is explained in depth, enriched by the author's extensive experience in research and teaching. Beyond classical methods, the book introduces innovative concepts, such as the Inverse Relationship Theorem and the alpha-method for tackling indeterminate systems, providing readers with powerful new tools for problem-solving. Designed for diverse learning environments, such as classroom instruction, online courses, or self-study, the book combines rigorous theory with practical applications. Python code examples are integrated throughout, allowing readers to see how mathematical concepts translate directly into formulas, numerical results, and visualizations. Readers are encouraged to experiment with the code to deepen understanding and develop their own solutions for related problems. Presented in Jupyter Notebook format, this book seamlessly unites theoretical explanations, mathematical formulations, and executable code in a single interactive document. This creates an engaging environment for learning, practicing, and exploring concepts in real time.

Heavy lifting in formula derivation and number crunching is handled by the computer, allowing readers to focus more on understanding the concepts.