Rank–Nullity Theorem Made Simple | With Full Verification

Master Rank, Null Space, and Rank–Nullity Theorem with an important VTU exam question, explained step-by-step. Perfect for BCS405D and 1BMATS101 students to understand core concepts of linear transformations. ━━━━━━━━━━━━━━━━━━━━━ 📌 Exact Question: ❓ Find the range, null space, rank, and nullity of the transformation T : V₃(R) → V₂(R) defined by T(x, y, z) = (y − x, y − z). Also verify the Rank–Nullity Theorem. ━━━━━━━━━━━━━━━━━━━━━ 🎯 Learning Points: ✔ Range (image) of transformation ✔ Null space (kernel) ✔ Rank and nullity calculation ✔ Rank–Nullity Theorem verification ✔ Matrix representation of transformation ✔ Exam-oriented solving method ━━━━━━━━━━━━━━━━━━━━━ 📚 Watch Next: 1️⃣ Rank & Nullity (3D to 3D) https://youtu.be/kDV_WUvL4gg 2️⃣ Rank & Nullity (Advanced) https://youtu.be/wkUEeHzdmbY 3️⃣ Full Playlist (Start from beginning) https://www.youtube.com/playlist?list=PLM9RnGtTy9_9lAWtTqWSNXi4ApcJUQx7k ━━━━━━━━━━━━━━━━━━━━━ 📘 Module Playlist (Start to Finish): ▶️ BCS405D – Module 2 (Linear Transformations) https://www.youtube.com/playlist?list=PLM9RnGtTy9_9lAWtTqWSNXi4ApcJUQx7k ▶️ 1BMATS101 – Module 5 (Linear Transformation) https://www.youtube.com/playlist?list=PLM9RnGtTy9__fllLHI_SsGvlsurgj-BjE ━━━━━━━━━━━━━━━━━━━━━ 📙 More Practice / Related Topics: 1️⃣ Vector Spaces https://www.youtube.com/playlist?list=PLM9RnGtTy9_9ybQUjtwVyix1Acy6B7eiG 2️⃣ Eigenvalues & Eigenvectors https://www.youtube.com/playlist?list=PLM9RnGtTy9_82B3CoOgAnhaV9V5mj2WI2 3️⃣ Inner Product Spaces https://www.youtube.com/playlist?list=PLM9RnGtTy9_-SUY_ZFZm_T2O9ofIOB3og ━━━━━━━━━━━━━━━━━━━━━ 🚀 Next Step: Move to rank-nullity in higher dimensions. 📌 Don’t stop here: Practice multiple rank-nullity problems. ━━━━━━━━━━━━━━━━━━━━━ 📘 Subject Details: 📚 Subject Codes: • BCS405D – Linear Algebra • 1BMATS101 – Calculus & Linear Algebra 📌 Topic: Linear Transformations ━━━━━━━━━━━━━━━━━━━━━ 📙 All Playlists (VTU Maths): ▶️ 1BMATS101 – Model Question Papers https://www.youtube.com/playlist?list=PLM9RnGtTy9_9LJxGYqFDpMOlhwYivpCAN ▶️ 1BMATS101 – Module 1 (Calculus) https://www.youtube.com/playlist?list=PLM9RnGtTy9_-rqPtcCwDq2MBKGy4pDk6K ▶️ 1BMATS101 – Module 2 (Vector Calculus) https://www.youtube.com/playlist?list=PLM9RnGtTy9_9CcDmlQM7n_q6ABPHD5As- ▶️ 1BMATS101 – Module 3 (Linear Algebra) https://www.youtube.com/playlist?list=PLM9RnGtTy9_-rSzBJZDttMpYRKQYzxTm4 ▶️ 1BMATS101 – Module 4 (Vector Space) https://www.youtube.com/playlist?list=PLM9RnGtTy9_88Z1aKoF-xcrhEZ3Z4ElAB ▶️ 1BMATS101 – Module 5 (Linear Transformation) https://www.youtube.com/playlist?list=PLM9RnGtTy9__fllLHI_SsGvlsurgj-BjE ━━━━━━━━━━━━━━━━━━━━━ ▶️ BCS405D – Module 1 (Vector Spaces) https://www.youtube.com/playlist?list=PLM9RnGtTy9_9ybQUjtwVyix1Acy6B7eiG ▶️ BCS405D – Module 2 (Linear Transformations) https://www.youtube.com/playlist?list=PLM9RnGtTy9_9lAWtTqWSNXi4ApcJUQx7k ▶️ BCS405D – Module 3 (Eigenvalues & Eigenvectors) https://www.youtube.com/playlist?list=PLM9RnGtTy9_82B3CoOgAnhaV9V5mj2WI2 ▶️ BCS405D – Module 4 (Inner Product Spaces) https://www.youtube.com/playlist?list=PLM9RnGtTy9_-SUY_ZFZm_T2O9ofIOB3og ▶️ BCS405D – Module 5 (Optimization Techniques) https://www.youtube.com/playlist?list=PLM9RnGtTy9__Xai2X0_I5mcu5v8jJkaqr ━━━━━━━━━━━━━━━━━━━━━ 👉 Follow VTU Maths with Muheeb (Mathematics Tutor) on WhatsApp https://whatsapp.com/channel/0029Vb6ct0L30LKOlXViVx3Y 👉 Get all VTU Maths updates and video links on Telegram https://t.me/vtumathswithmathematicstutor 💎 Support Us: Join our channel and get access to exclusive perks 👇 https://www.youtube.com/channel/UC_OTtwh8_-q-bA7DoS--syg/join ━━━━━━━━━━━━━━━━━━━━━ #LinearTransformation #RankNullity #LinearAlgebra #VTUMaths #BCS405D #1BMATS101 #EngineeringMaths