Linear Algebra Part 2: Vector Spaces, Eigenvalues, and Applications

Linear algebra is the mathematical engine driving modern technology, from computer graphics to machine learning algorithms. This course picks up where basic matrix arithmetic ends, guiding you through the core structural concepts of advanced mathematics. You will transition from simply computing numbers to deeply understanding the geometric and algebraic structures of vector spaces, linear transformations, and matrix decompositions. What you'll learn: Understand the formal definitions of vector spaces, subspaces, span, and linear independence; Map transformations between spaces and analyze their kernel, image, and rank; Master eigenvalues and eigenvectors to understand matrix behavior and system stability; Explore inner product spaces, orthogonality, and the Gram-Schmidt orthogonalization process; Examine modern applications of matrix decompositions, including foundational concepts of Singular Value Decomposition (SVD) used in data analysis. This written course begins with essential terminology and foundational definitions before moving into structured proofs, clear step-by-step derivations, and practical written exercises. Designed for beginners who have a basic familiarity with matrices, this guide requires no advanced mathematical prerequisites. Start reading today to build a rigorous mathematical foundation for your technical career.