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Quantum Engineering Year 0: Mathematical Foundations Month 4Day 89

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Year 0·Month 4·Week 1

Day 89: Bases and Dimension

Day 89 of 2,016~15 min read

Learning Objectives

•Define basis and state its two essential properties
•Prove that every basis of a finite-dimensional space has the same size
•Calculate the dimension of vector spaces and subspaces
•Find bases for various vector spaces
•Understand coordinate representations
•Connect bases to complete sets of states in quantum mechanics

Today's Schedule (7 hours)

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1 Definition of Basis 2 Equivalent Characterization 3 Standard Bases Standard Basis of and Standard Basis of mathcalP_nmathbbF Standard Basis of M_m times nmathbbF 4 Finite-Dimensional Vector Spaces 5 The Dimension Theorem 6 Key Theorems About Dimension 7 Finding Bases For subspaces defined by equations For spans 8 Coordinate Vectors Quantum Mechanics Connection Complete Orthonormal Bases Dimension and Quantum Systems The Computational Basis Change of Basis in QM Dimension Determines Information Content

Day 88Day 89 of 2,016Day 90