Linear Algebra: Vector Spaces and Matrix Decompositions

Linear algebra is the mathematical engine driving modern technology, from computer graphics to machine learning algorithms. To truly understand how complex data is structured and manipulated, mastering vector spaces and linear transformations is essential. This text-only course guides you step-by-step through the core concepts of advanced linear algebra, translating abstract mathematical theory into clear, intuitive explanations. You will build a strong mathematical foundation and learn how these principles apply to modern computational fields. What you'll learn: • Understand the formal definitions and properties of vector spaces, subspaces, and bases. • Master linear transformations, kernel, image, and matrix representations. • Compute and interpret eigenvalues, eigenvectors, and diagonalization processes. • Explore inner product spaces, orthogonality, and the Gram-Schmidt orthogonalization process. • Apply singular value decomposition (SVD) and principal component analysis (PCA) concepts to data structures. • Practice solving theoretical proofs and computational problems through structured written exercises. This course begins with essential terminology and foundational definitions before moving into abstract vector spaces, linear mappings, and advanced matrix decompositions. Each concept is explained with detailed written breakdowns and step-by-step mathematical proofs. This course is designed for students, self-taught programmers, and aspiring data scientists who have a basic grasp of high school algebra and want to master the mathematical foundations of modern computing. No advanced mathematical background is required. Start reading today to unlock the mathematical principles that power modern algorithms.