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Quantum Engineering Year 0: Mathematical Foundations Month 4Day 93

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Year 0·Month 4·Week 2

Day 93: Matrix Representation of Linear Maps

Day 93 of 2,016~16 min read

Learning Objectives

•Construct the matrix of a linear transformation
•Understand how the matrix depends on choice of bases
•Apply matrix multiplication to compute T(v)
•Convert between linear maps and matrices fluently
•Represent quantum operators as matrices in different bases

Today's Schedule (7 hours)

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Schedule Overview Learning Objectives Required Reading Morning Session Theory 35 hours Part 1 The Matrix of a Linear Map 75 min The Fundamental Construction Why This Definition Notation Example 1 Rotation in Example 2 Derivative as a Linear Map Part 2 Matrices Depend on Bases 45 min A Critical Point Example Reflection Across y x Change of Basis Matrix Part 3 Matrix-Vector Multiplication 30 min Definition Revisited Column Picture Row Picture Part 4 The Vector Space of Matrices 30 min M_mnF is a Vector Space Connection to Linear Maps Part 5 Quantum Connection Operators as Matrices 30 min Quantum Operators in Computational Basis Pauli Matrices Hadamard Gate Operator in Different Bases Afternoon Session Problem Solving 25 hours Problem Set A Matrix Construction 50 min Problem Set B Change of Basis 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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