Vector Space & Subspace | Linear Algebra | Lec -1 | CSIR NET MATHS | IFAS

Vector Space & Subspace | Linear Algebra | Lec -1 | CSIR NET MATHS explained with clear concepts and exam-focused approach. Master Vector Space & Subspace fundamentals for CSIR NET Maths with IFAS. Includes definitions, one-step test, PYQ-based examples, and problem-solving tricks for high scores. 🔥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: Vector Space & Subspace [00:07:32] – Important: Vector Space Definition [00:08:11] – Concept: Internal & External Composition [00:10:42] – Axioms: Abelian Group & Distributive Laws [00:16:11] – Important: Field over Subfield is Vector Space [00:18:24] – Subspace Definition [00:20:01] – Important: Zero Vector & Improper Subspace [00:21:02] – Question 1: Subspaces of R² over Q² [00:25:34] – Important: One-Step Subspace Test [00:27:04] – Function Space: Parent Vector Space [00:30:44] – Pointwise Addition & Scalar Multiplication [00:31:33] – Question 2: Subspace of Function Space [00:34:46] – PYQ: Subspace Verification (4 Options) [00:38:09] – Trick: Sum of Functions Fails Subspace [00:40:03] – Fixed Point Function Analysis [00:43:37] – Question 3: Final Subspace Example [00:48:04] – Important: Subset Failure via Domain [00:51:05] – Zero at Fixed Point: Subspace Passes [00:53:09] – Non-Negative Function Fails Subspace [00:55:03] – Complete Session: (Vector Space & Subspace Lec-1) ━━━━━━━━━━━━━━━━━━━━━━━━━ 📖 Topics Covered in This Lecture: • Vector Space Definition: Axioms & Field Over Subfield Concepts • Subspace Definition: Proper & Improper Subspaces Explained • One-Step Test: Subspace Verification Shortcut & PYQs • Function Space: Set of All Functions as Vector Space • Point-wise Addition & Scalar Multiplication Concepts • Standard Examples: R over Q, C over R & Finite Fields • Subspace PYQs: Checking Subset, Closure & Scalar Multiplication ━━━━━━━━━━━━━━━━━━━━━━━━━ 📚 Lecture Summary: In this session, IFAS Expert Faculty focuses on Vector Space & Subspace for CSIR NET Maths. Covers the formal definition, field over subfield concept, function space, and the one-step subspace test. Includes Vector Space & Subspace PYQ solving, proper vs improper subspace identification, and point-wise operation verification for exam success. ━━━━━━━━━━━━━━━━━━━━━━━━━ 🎓 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   #linearalgebra #vectorspace #csirnetmathematicalscience #gatemathematics #csirnetmaths #csirnetdec2026 #csirnetlecture #ifasmathematics