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Quantum Engineering Year 0: Mathematical Foundations Month 4Day 96

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Year 0·Month 4·Week 2

Day 96: The Rank-Nullity Theorem

Day 96 of 2,016~16 min read

Learning Objectives

•State and prove the rank-nullity theorem
•Apply it to determine dimensions of kernel and range
•Use it to analyze solvability of linear systems
•Understand its geometric meaning
•Connect to quantum state space dimensions

Today's Schedule (7 hours)

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Schedule Overview Learning Objectives Required Reading Morning Session Theory 35 hours Part 1 Statement of the Theorem 30 min The Fundamental Theorem of Linear Maps Why This is Profound Matrix Form Part 2 The Proof 60 min Setup Claim Tv Tv is a basis for rangeT Conclusion Part 3 Applications 60 min Application 1 Injectivity and Dimension Application 2 Surjectivity and Dimension Application 3 Isomorphism Criterion Application 4 Number of Solutions to Ax b Application 5 Counting Free Variables Part 4 Geometric Interpretation 30 min Dimension as Degrees of Freedom Visual Picture Part 5 Quantum Connection 30 min Dimension Conservation in QM Example Spin-12 System Afternoon Session Problem Solving 25 hours Problem Set A Basic Applications 50 min Problem Set B Matrix Problems 50 min Problem Set C Proofs 50 min Evening Session Computational Lab 1 hour Homework Written Problems Daily Checklist Preview Tomorrow

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