Vector Space And Subspace | JRF EXPRESS | Target CSIR NET June 2026 | Lec-01 | IFAS

Vector Space And Subspace explained with clear concepts and exam-focused approach. This JRF Express session targets CSIR NET June 2026 with a rigorous practice of previous year questions. Dive deep into subspace dimension tricks, linear dependence tests, and finite field vector space counting problems. With step‑by‑step problem‑solving, this lecture strengthens your core linear algebra concepts, ensuring you confidently handle any subspace or dimension question in the exam. 🔥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 Vector Space And Subspace [00:03:38] – Question 1: Recurrence subspace dimension [00:08:29] – Question 2: Subspaces containing \(W\) (kernel) [00:16:17] – Question 3: Rank & LI of \(Me_i\) [00:20:13] – Question 4: F₃ subspace counts & maps [00:22:06] – Formula: Number of \(k\)-dim subspaces (F₃) [00:29:11] – Question 5: Polynomial subspace condition [00:32:38] – Question 6: \(W_1\cap W_2\) dimension [00:34:53] – Trick: Zero‑position counting (inclusion–exclusion) [00:35:58] – Question 7: LI/LD statements in \(\mathbb{R}^n\) [00:39:12] – Question 8: Dual of span\(\{e^{t},e^{2t},e^{3t}\}\) [00:41:02] – Concept: \(\dim L(V,W)=\dim V\cdot\dim W\) [00:43:56] – Complete Session (Vector Space And Subspace ) ━━━━━━━━━━━━━━━━━━━━━━━━━ 📖 Topics Covered in This Lecture: • Vector Space & Subspace Definition and Test PYQs • Dimension Calculation: Independent Entries & Restrictions Tricks • Linear Dependence / Independence Concepts & Counterexamples • Finite Field Vector Spaces: Subspace Counting & Cardinality • Span, Basis, and Dimension for Polynomial & Function Spaces • Intersection of Subspaces & Dimension Formula • Linear Transformations: Rank-Nullity & Kernel Dimension ━━━━━━━━━━━━━━━━━━━━━━━━━ 📚 Lecture Summary: In this session, IFAS Expert Faculty focuses on vector space and subspace problems from the JRF Express series for CSIR NET June 2026. Key concepts include determining dimension by counting free parameters, applying subspace test conditions, and using the union of restrictions for subspace intersections. The problem‑solving approach covers finite field subspace counting, identifying linear dependence, and computing dimensions of spaces of linear transformations. The vector space and subspace techniques are reinforced through direct PYQs, making this an essential revision for mastering linear algebra. ━━━━━━━━━━━━━━━━━━━━━━━━━ 🎓 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   #vectorspaceandsubspace #jrfexpress #csirnetjune2026 #linearalgebra #subspacedimension #lineardependence #finitefieldvectorspaces #dimensioncalculation #csirnetmathematicalscience #gatemathematics #csirnetmaths #csirnetjune2026 #csirnetlecture #ifasmathematics