Linear Algebra Part II: Vector Spaces and Transformations

Transitioning from basic matrix arithmetic to abstract vector spaces is a major milestone in mathematics. This text-based course guides you through intermediate and advanced linear algebra, bridging abstract theory with rigorous problem-solving and modern applications. You will learn to: Understand vector spaces, subspaces, and the principles of linear independence and bases; Analyze linear transformations, their matrix representations, and kernel and image spaces; Compute eigenvalues, eigenvectors, and diagonalize matrices for theoretical and practical use; Explore inner product spaces, orthogonality, and the Gram-Schmidt process; Apply linear algebra foundations to modern computational concepts like PCA and data dimensionality reduction; Practice solving challenging proofs and exam-style problems. We begin with key terminology and foundational definitions of vector spaces before advancing to transformations, inner products, and exam-focused practice. This course is designed for mathematics students, competitive exam aspirants such as those preparing for IIT JAM, and learners seeking a deep mathematical foundation for data science. Start reading today to master advanced linear algebra.