📘 Master Course in Pure Mathematics: Linear Algebra, Real Analysis & Complex Analysis

Course Overview:

This intensive course offers a rigorous and comprehensive journey through the foundational pillars of pure mathematics — Linear Algebra, Real Analysis, and Complex Analysis. Designed for advanced undergraduates, graduate students, and competitive exam aspirants, the course focuses on deep conceptual understanding, formal proofs, and challenging problem-solving.

🧮 Module 1: Linear Algebra (The Language of Structure)

• Vector spaces, subspaces, linear dependence and independence
• Bases, dimension, and coordinate systems
• Linear transformations and matrix representations
• Rank-nullity theorem, fundamental isomorphism theorems
• Inner product spaces, orthogonality, Gram-Schmidt process
• Eigenvalues, eigenvectors, diagonalization
• Minimal and characteristic polynomials, Cayley-Hamilton theorem
• Jordan canonical form, nilpotent operators, applications
• Quadratic Form and Inner product space

Focus: Proof-based understanding and transformation theory with challenging matrix problems and canonical forms.

---

📏 Module 2: Real Analysis (The Heart of Rigour)

• Properties of the real number system, completeness axiom
• Sequences and series, convergence criteria
• Limits, continuity, uniform continuity
• Differentiability and the Mean Value Theorem
• Riemann integration and its properties
• Sequences and series of functions, uniform convergence
• Function of several variables

Focus: Developing analytical maturity through ε-δ proofs, deep theorems, and counterexamples. Emphasis on constructing rigorous arguments and exploring pathological cases.

---

🔍 Module 3: Complex Analysis (The Power of Analyticity)

• Complex numbers, topology of ℂ, open/closed/connected sets
• Analytic functions, Cauchy-Riemann equations
• Contour integration, Cauchy’s Theorem and Integral Formula
• Liouville’s Theorem, Maximum Modulus Principle
• Power series, Laurent series, classification of singularities
• Residue calculus and evaluation of integrals
• Argument principle, Rouche's Theorem, conformal mappings
• Bilinear Transformation

Focus: Visual intuition meets formal rigor. Strong emphasis on classical results and problem-solving techniques in integration and series.

---

🎯 Course Features:

• 250+ hours of live & recorded lectures
• Daily practice sheets & advanced assignments
• Topic-wise PYQ (CSIR-NET & GATE) discussion
• Weekly tests & detailed solution sessions
• Special focus on proof techniques & counterexample construction
• Personalized doubt sessions and mentorship support

---

🧠 Ideal For:

• CSIR-NET, GATE, TIFR, NBHM, ISI, Putnam aspirants
• B.Sc/M.Sc Math students aiming for mastery
• Aspiring researchers in mathematics or theoretical sciences
• Anyone who wants to "think like a mathematician"