Linear Algebra & Complex Analysis | Target CSIR NET June 2026 | Lec-1 | IFAS

Linear Algebra & Complex Analysis | Target CSIR NET June 2026 | Lec-1 | IFAS explained with clear concepts and exam-focused approach. Master linear algebra, complex analysis PYQs, and problem‑solving tricks for CSIR NET June 2026. Covers rank, nilpotent matrices, analytic functions, residues, and more.

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⏳ Lecture Timeline:
[00:00:00] – Introduction to Linear Algebra & Complex Analysis
[00:03:42] – Question 1: Rank of Matrix Products
[00:08:40] – Question 2: Nilpotent Matrix (P⁴=0)
[00:11:02] – Trick: Nilpotent ⇒ All Eigenvalues 0
[00:12:37] – Concept: Jordan Block & Rank of P^k
[00:14:54] – Question 2 (contd.): Similarity & JC Form
[00:16:03] – Question 3: 1‑1/Onto Linear Transformation
[00:20:01] – Question 4: R over Q vs Q[x] Dimension
[00:22:52] – Trick: Uncountable vs Countable Dimension
[00:30:12] – Question 5: Null Space of m×n Matrices
[00:33:20] – Trick: Nullity ≥ 3 for 2×5 Matrix
[00:36:29] – Question 6: Quadratic Form Non‑Trivial Zero
[00:38:36] – Trick: Direct Substitution to Check Zero
[00:41:52] – Question 7: Diagonalisability (Permutation Matrices)
[00:44:57] – Concept: Diagonalisable over C vs R
[00:46:20] – Question 8: Polynomial Subspaces Dimension
[00:49:42] – Trick: W₁+W₂ Dimension Formula
[00:56:14] – Complex Analysis Start
[01:02:03] – Question 9: Analytic Function f(z)=ze^z
[01:03:48] – Trick: Separate Real & Imaginary Parts
[01:06:19] – Question 10: sinz/z Entire Function
[01:11:17] – Question 11: Cauchy Integral Consequences
[01:16:35] – Important: Entire Bounded ⇒ Polynomial
[01:20:18] – Question 12: Conditions ⇒ Constant Function
[01:24:56] – Question 13: Residues of 1/(z(z‑1))
[01:34:18] – Question 14: sin(1/z) Essential Singularity
[01:36:04] – Concept: Casorati‑Weierstrass & Picard
[01:37:29] – Question 15: Argument Principle & Zeros
[01:41:09] – Question 16: Entire Function Properties
[01:45:45] – Question 17: (e^z‑1)/z Singularity
[01:51:18] – Homework & Session End
[01:52:28] – Complete Session (Linear & Complex PYQs)

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📖 Topics Covered in This Lecture:
• Matrix Rank & Product Rank Bounds PYQs
• Nilpotent Matrix & Jordan Form Tricks
• Linear Transformation (1‑1, Onto) Problems
• Quadratic Form Non‑Trivial Zero Concepts
• Diagonalizability Over R and C PYQs
• Analytic Function, Harmonic Conjugate & Cauchy Integral
• Residue Theorem, Essential Singularity & Picard Theorem

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📚 Lecture Summary:
In this session, IFAS Expert Faculty focuses on linear algebra and complex analysis with a problem‑solving approach for CSIR NET June 2026. Key topics: rank of product matrices, nilpotent matrices & Jordan blocks, linear transformations, quadratic forms, diagonalizability over R and C, analytic functions, harmonic conjugates, residues, and essential singularities. PYQs and tricks reinforce both linear algebra and complex analysis concepts effectively.

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🎓 This lecture is highly beneficial for students preparing for:
✅ CSIR NET Mathematical Science ✅ GATE Mathematics ✅ SET Mathematics
✅ TIFR Mathematics ✅ PhD Entrance Exams

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✅ CSIR NET Dec 2025: 4000+ Selections (AIR 3, 8, 9.....)
✅ CSIR NET June 2025: 4200+ Selections (AIR 1, 2, 6, 10...)
✅ CSIR NET Dec 2024: 4000+ Selections (AIR 5, 6, 7, 8, 9...)
✅ CSIR NET June 2024: 2000+ Selections (AIR 3, 10, 13...)
✅ CSIR NET Dec 2023: 1200+ Selections (AIR 2, 4, 5...)

📄 View IFAS Result Sheet:
👉 https://drive.google.com/file/d/1THh6TuLtCuSyAYczcHygIV22F87jD238/view

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