Linear Algebra ll | One Short Series | CSIR NET July 2026 | Lec- 8| IFAS

Linear Algebra ll | One Short Series | CSIR NET July 2026 | Lec- 8| IFAS explained with clear concepts and exam-focused approach. This session covers advanced Linear Algebra, including eigenvalues, minimal polynomial, Jordan canonical form, and quadratic forms with high‑yield PYQ tricks for CSIR NET, GATE, and SET. 🔥Join India's Best Coaching for CSIR NET and GATE Exams: https://ifasonline.com/csir-net/mathe... 📞 Need Guidance? Call/WhatsApp: 9172266888 ━━━━━━━━━━━━━━━━━━━━━━━━━ ⏳ Lecture Timeline: [00:00:00] – Introduction of Linear Algebra ll [00:04:19] – Question 1: LI of T1,T2 from T1±T2 [00:09:42] – Important: dim L(V) = n², Extending LI Sets [00:13:08] – Question 2: Annihilating Polynomial, Minimal Poly [00:15:44] – Important: Scalar Operator T=2I → Commutes with All [00:19:31] – Question 3: Bilinear Form, Indefinite Quadratic [00:24:30] – Important: Indefinite ⇒ Range is Whole ℝ [00:27:09] – Question 4: Span of I,A,…,Aⁿ, Dimension [00:28:56] – Concept: Dim of Span = Degree of Minimal Poly [00:50:02] – Question 6: Primary Decomposition, Irreducible Factors [00:52:45] – Concept: Primary Decomposition Theorem [01:06:59] – Question 7: 5×5 Matrix with Given Minimal Poly [01:16:10] – Question 8: Similarity from Minimal Poly, n & Eigenvalues [01:44:13] – Question 9: Symmetric Bilinear Form, 2‑Dim [01:53:30] – Question 10: Dual Space Dimension, W Subspace [01:56:21] – Question 11: Skew‑Symmetric Bilinear Form, Det [02:02:03] – Skew Matrix, Determinant = 1, Non‑Degenerate [02:09:02] – Question 12: Permutation Matrices Diagonalizability [02:12:47] – Question 13: Inner Product, Basis S in C[0,π] [02:22:15] – Question 14: 10×10 Rank 7, Generalized Eigenvectors [02:27:07] – All Zero Eigenvalues Possible (Nilpotent) [02:31:05] – Question 15: Symmetric Bilinear Form, Nullspace [02:36:05] – Question 16: Quadratic Form with Non‑Zero Zero [02:38:11] – Completing Square to Check Definiteness [02:42:16] – Complete Session (Linear Algebra PYQs) ━━━━━━━━━━━━━━━━━━━━━━━━━ 📖 Topics Covered in This Lecture: • Eigenvalues, Minimal Polynomial & Jordan Canonical Form • Similarity of Matrices & Diagonalization Conditions • Quadratic Forms: Definiteness, Degenerate & Non‑Degenerate • Symmetric & Skew‑Symmetric Bilinear Forms • Inner Product Spaces, Orthonormal Basis & Subspaces • Linear Transformations, Rank‑Nullity & Dimension Tricks ━━━━━━━━━━━━━━━━━━━━━━━━━ 📚 Lecture Summary: In this session, IFAS Expert Faculty focuses on Linear Algebra with a one‑shot revision of high‑level topics for CSIR NET July 2026. Key concepts include minimal polynomial and characteristic polynomial relationships, Jordan canonical form and similarity of complex matrices, classification of quadratic forms via matrix representation, and properties of symmetric bilinear forms. Problem‑solving approaches using dimension arguments, rank‑nullity, and generating sets are covered with PYQ relevance, making this Linear Algebra session crucial for final exam readiness. ━━━━━━━━━━━━━━━━━━━━━━━━━ 🎓 This lecture is highly beneficial for students preparing for: ✅ CSIR NET Mathematical Science ✅ GATE Mathematics ✅ SET Mathematics ✅ TIFR Mathematics ✅ PhD Entrance Exams ━━━━━━━━━━━━━━━━━━━━━━━━━ 🏆 Why IFAS is India’s No. 1 Choice for CSIR NET & GATE Mathematics? (Our Results Speak): ✅ 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/1THh6... ━━━━━━━━━━━━━━━━━━━━━━━━━ 🔗 Important Links: 🌍 Visit our Website: https://ifasonline.com 📞 Contact for Queries: 9172266888 📘 Free Counselling: https://ifasonline.com/csir-net/mathe... ━━━━━━━━━━━━━━━━━━━━━━━━━ 🔗 India's Best Online Classes and Books: 🔥CSIR NET Mathematical Science Live & Recorded Course: https://ifasonline.com/csir-net/mathe... 🎯 GATE Mathematics Recorded Course: https://ifasonline.com/gate/mathemati... 📘CSIR NET Mathematical Science Books: https://ifasonline.com/csir-net/mathe... ━━━━━━━━━━━━━━━━━━━━━━━━━ 📲 Download IFAS App: Android: https://play.google.com/store/apps/de... IOS: https://apps.apple.com/us/app/ifas/id... ━━━━━━━━━━━━━━━━━━━━━━━━━ Stay Updated Via Other Social Media: ✔Telegram: https://t.me/NET_GATE_SET_Maths_Discu... ✔Instagram:   / csir_net_mathematical_science   #linearlalgebraii #linearalgebra #oneshotseries #csirnetmathematicalscience #gatemathematics #csirnetmaths #csirnetjune2026 #csirnetlecture #ifasmathematics