Linear Algebra II: Vector Spaces and Linear Transformations for IIT JAM

To succeed in advanced mathematics and competitive exams like the IIT JAM, a deep understanding of core algebraic structures is essential. This text-based course builds directly on foundational matrix algebra to elevate your mathematical reasoning and analytical skills. You will progress from basic matrix operations to mastering abstract vector spaces, linear transformations, and eigenvalues, developing the rigorous problem-solving techniques required for high-stakes exams. What you'll learn: - Understand the formal structure of vector spaces, subspaces, and direct sums. - Analyze linear transformations, their kernel, image, and rank-nullity theorem applications. - Compute eigenvalues, eigenvectors, and determine matrix diagonalizability. - Apply inner product space concepts, including orthogonality and the Gram-Schmidt process. - Practice solving complex, exam-style problems with step-by-step written derivations. - Connect abstract algebraic concepts to modern applications in data science and computational algorithms. The course begins with foundational definitions of abstract vector spaces before moving systematically through linear maps, matrices of transformations, and inner product spaces. You will read detailed explanations and work through structured mathematical exercises designed to build exam readiness. Designed for undergraduate mathematics students and competitive exam aspirants, particularly those preparing for the IIT JAM, who have a basic familiarity with introductory matrix operations. Read, practice, and master the core principles of advanced linear algebra today.