Foundations of Linear Algebra and Ordinary Differential Equations

Mathematical modeling is the backbone of modern engineering, physics, and data science. To analyze dynamic systems and work with complex datasets, you must first build a strong intuitive understanding of how linear algebra and differential equations interact. This course bridges the gap between abstract mathematical theory and practical application. You will transition from calculating basic matrix operations to solving complex systems of differential equations. By learning how to frame physical systems as mathematical models, you will gain the analytical skills needed to interpret dynamic behaviors in engineering, economics, and physical sciences. What you'll learn: - Understand foundational definitions of vector spaces, subspaces, and linear independence - Solve first-order ordinary differential equations using separable and linear methods - Calculate eigenvalues and eigenvectors to analyze matrix transformations and stability - Apply matrix algebra to solve systems of linear algebraic equations efficiently - Model dynamic systems using coupled first-order linear differential equations - Interpret the behavior and stability of solutions to higher-order linear equations This text-based course begins with the essential definitions of linear algebra, introducing vectors, matrices, and system solutions. Next, you will explore first-order differential equations before combining both disciplines to solve multi-variable systems of equations with step-by-step written explanations and practical exercises. This course is designed for beginners, students, and self-taught professionals looking to build a robust mathematical foundation. No prior experience with differential equations or advanced matrix theory is required, though basic calculus is recommended. Start your journey into advanced mathematical modeling and master the tools of modern engineering analysis today.