Linear Transformations for Mathematics Entrance Exams

Preparing for higher-level mathematics entrance exams requires a deep, intuitive understanding of linear algebra. This written course guides you through the core principles of linear transformations, helping you build the problem-solving speed and analytical accuracy needed for competitive exams. You will transition from basic vector space concepts to mastering complex mapping, matrix representations, and spectral theory. Through clear written explanations and step-by-step proofs, you will develop the skills to tackle rigorous exam questions with confidence. What you'll learn: Understand the foundational properties of vector spaces, subspaces, and bases; Define and analyze linear transformations, kernels, and ranges; Apply the Rank-Nullity Theorem to solve dimension-related exam problems; Construct matrix representations of linear maps under different bases; Calculate eigenvalues, eigenvectors, and diagonalize linear operators; Practice solving typical postgraduate entrance exam questions using structured written proofs. The course begins with essential definitions of vector spaces before moving systematically through linear mappings, matrix representations, and diagonalizability. Each module features clear theoretical breakdowns followed by written practice problems designed to mirror exam formats. This course is designed for undergraduate students preparing for postgraduate mathematics entrance exams, as well as anyone seeking a rigorous refresher in linear algebra. No advanced prerequisites are required, as we build up from foundational definitions. Start reading today to master linear transformations and elevate your exam preparation.