Algebra of Subspaces | Linear Alegbra | CSIR NET MATHEMATICS | Lec-04 | IFAS

Algebra of Subspaces | Linear Alegbra | CSIR NET MATHEMATICS explained with clear concepts and exam-focused approach. Master union, intersection, and dimension formulas for CSIR NET Mathematics with PYQs, proofs, and problem-solving tricks for subspaces. 🔥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: Algebra of Subspaces [00:02:19] – Result Predictor & Batch Info [00:03:31] – PYQ: Union of Proper Subspaces [00:05:13] – Important: Union of Two Subspaces Never Equals V [00:06:44] – Case Analysis: W1 and W2 Distinct [00:10:32] – Important: Result Holds for Any Field [00:11:23] – PYQ: Union of Three Subspaces [00:13:17] – Concept: Subgroups vs Subspaces [00:16:03] – Important: Three Subspaces Can Equal V [00:16:25] – PYQ: Unique Basis Condition [00:17:23] – Trick: One-Dimensional Space Logic [00:20:01] – Example: Z2 over Z2 for Unique Basis [00:21:04] – PYQ: Precisely Two Bases [00:22:09] – Concept: Finite Field Cardinality [00:24:03] – Important: p^k Elements in Finite Field [00:26:52] – Algebra of Subspaces: Intersection [00:27:25] – Important: Union Condition (Containment) [00:28:11] – Concept: Sum of Subspaces [00:29:50] – Important: Dimension Formula for W1+W2 [00:30:09] – PYQ: Intersection of Trace-Zero & Symmetric [00:33:07] – Trick: LI Restrictions Method [00:34:43] – PYQ: Polynomial Spaces W1 & W2 [00:37:26] – W1 Dimension: 21-4 = 17 [00:41:28] – W1 ∩ W2 Dimension: 21-6 = 15 [00:43:28] – W1 + W2 Dimension: 17+17-15 = 19 [00:45:22] – Concept: Dimension over Subfield [00:48:02] – Important: Tower Law (Dimension Formula) [00:48:52] – Note: Holds for Infinite Dimensions [00:49:06] – Homework: Tower Law PYQ [00:49:21] – Complete Session: (Algebra of Subspaces) ━━━━━━━━━━━━━━━━━━━━━━━━━ 📖 Topics Covered in This Lecture: • Union of Subspaces: Proof & Counter-Examples for Proper Subspaces • Intersection & Sum of Subspaces: Dimension Formulas & LI Restrictions • Finite Dimensional Spaces: Unique Basis & Cardinality Concepts • Matrix Subspaces: Trace Zero, Symmetric Matrices & Dimension Calculation • Field Extensions: Tower Law & Vector Space Dimension over Subfield ━━━━━━━━━━━━━━━━━━━━━━━━━ 📚 Lecture Summary: In this session, IFAS Expert Faculty focuses on Algebra of Subspaces in Linear Algebra for CSIR NET Mathematics. Covers union/intersection theorems, dimension formulas, and finite field examples. Includes proofs, PYQs, and shortcut tricks for calculating dimensions using linearly independent restrictions. Essential for mastering subspace properties and scoring high in the exam. ━━━━━━━━━━━━━━━━━━━━━━━━━ 🎓 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 #algebraofsubspaces #csirnetmathematicalscience #gatemathematics #csirnetmaths #csirnetdec2026 #csirnetlecture #ifasmathematics