Linear Algebra Foundations: Orthogonal Diagonalization and SVD

Modern data science, machine learning, and image processing rely heavily on structural matrix decompositions to simplify complex datasets. Understanding the underlying linear algebra is essential for anyone wanting to write efficient algorithms or grasp how modern AI models process multidimensional data. This text-based course guides you from fundamental vector spaces to advanced matrix factorization techniques. You will build a strong intuitive and mathematical foundation in matrix operations, transitioning from basic geometric transformations to key decomposition theorems. By studying clear written explanations and examining practical Python code implementations, you will learn how to decompose matrices to extract their most important features. What you'll learn: - Understand the core principles of eigenvalues, eigenvectors, and symmetric matrices - Master orthogonal diagonalization and its geometric interpretation in vector spaces - Apply Singular Value Decomposition (SVD) to factorize any real matrix - Implement matrix decomposition algorithms in Python using modern NumPy conventions - Practice dimensionality reduction techniques such as Principal Component Analysis (PCA) - Explore how matrix factorization powers recommendation systems and image compression We begin with essential terminology, symmetric matrix properties, and inner product spaces. From there, you will progress to orthogonal projections, spectral theorem applications, and the step-by-step mechanics of SVD. The course concludes with practical, text-based coding exercises that show you how to apply these mathematical tools to real-world data problems. This course is designed for beginner data scientists, software engineers, and students who have a basic familiarity with algebra and Python but want to master the mathematical foundations of machine learning. No advanced linear algebra background is required. Start reading today to unlock the mathematical foundations of modern data algorithms.