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Quantum Engineering Year 0: Mathematical Foundations Month 4Day 92

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Year 0·Month 4·Week 2

Day 92: Definition of Linear Transformations

Day 92 of 2,016~17 min read

Learning Objectives

•Define linear transformations precisely
•Verify whether a function is linear
•Understand the structure-preserving nature of linear maps
•Work with standard examples (rotation, reflection, projection)
•Prove basic properties of linear transformations
•Connect linear maps to quantum operators

Today's Schedule (7 hours)

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Schedule Overview Learning Objectives Required Reading Morning Session Theory 35 hours Part 1 The Definition 60 min Motivation Structure-Preserving Functions Definition Linear Transformation Linear Map Why Linear Notation and Terminology Part 2 Fundamental Properties 45 min Theorem T0 0 Theorem T-v -Tv Theorem Linear Maps Preserve Linear Combinations Key Consequence Determined by Action on Basis Part 3 Standard Examples 60 min Example 1 The Zero Transformation Example 2 The Identity Transformation Example 3 Scalar Multiplication Map Example 4 Rotation in by Angle Example 5 Projection onto a Line Example 6 Differentiation Example 7 Integration Part 4 Non-Examples 30 min Non-Example 1 fx x 1 Non-Example 2 fx x Non-Example 3 fx y x y Non-Example 4 fx y xy Afternoon Session Problem Solving 25 hours Problem Set A Verification 45 min Problem Set B Construction 45 min Problem Set C Proofs 60 min Evening Session Computational Exploration 1 hour Homework Written Problems Complete before tomorrow Coding Assignment Daily Checklist Preview Tomorrow Key Insight of the Day

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