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Succeeding in competitive mathematics exams like the IIT JAM requires more than memorizing formulas; it demands a deep, intuitive understanding of core algebraic structures. This text-based course is designed to transition you from basic matrix operations to abstract vector spaces, providing the rigorous conceptual foundation and analytical skills necessary for undergraduate exams. You will learn to dissect complex mathematical arguments and solve exam-style problems with precision and confidence. What you'll learn: - Understand vector spaces, subspaces, and the concepts of linear independence and basis. - Apply linear transformations, find their matrix representations, and calculate rank and nullity. - Compute eigenvalues and eigenvectors, and determine the diagonalizability of a matrix. - Solve systems of linear equations using Gaussian elimination and matrix decompositions. - Master inner product spaces, orthogonality, and the Gram-Schmidt orthogonalization process. - Practice rigorous mathematical proofs and deductive reasoning for exam-level problem solving. This course begins with key terminology, basic definitions, and fundamental matrix algebra before progressing to advanced abstract vector spaces and linear mappings. Through clear written explanations, step-by-step proofs, and targeted practice problems, you will build a robust mathematical toolkit. This course is designed for undergraduate mathematics students and aspirants of competitive entrance exams, requiring only a basic familiarity with high school algebra. Start mastering linear algebra today and elevate your exam preparation.