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Quantum Engineering Year 0: Mathematical Foundations Month 4Day 95

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Year 0·Month 4·Week 2

Day 95: Kernel and Range

Day 95 of 2,016~17 min read

Learning Objectives

•Define and compute the kernel (null space) of a linear map
•Define and compute the range (image) of a linear map
•Prove both are subspaces
•Find bases for kernel and range
•Understand the connection to solving linear systems
•Connect to quantum measurement (projection operators)

Today's Schedule (7 hours)

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Schedule Overview Learning Objectives Required Reading Morning Session Theory 35 hours Part 1 The Kernel Null Space 60 min Definition Theorem The Kernel is a Subspace Nullity Finding the Kernel Example 1 Kernel of a 34 Matrix Part 2 The Range Image 60 min Definition Theorem The Range is a Subspace Rank Finding the Range Example 2 Range of the Same Matrix Part 3 Injectivity and Surjectivity 45 min Injective One-to-One Surjective Onto Bijective Invertible Part 4 Connection to Linear Systems 30 min Solving Ax b Summary Table Part 5 Quantum Connection Projections 30 min Projection Operators Key Properties of Projections Quantum Measurement Example Spin Measurement Afternoon Session Problem Solving 25 hours Problem Set A Kernel Computation 50 min Problem Set B Range Computation 50 min Problem Set C Proofs 50 min Evening Session Computational Lab 1 hour Homework Written Problems Daily Checklist Preview Tomorrow

Day 94Day 95 of 2,016Day 96