PU-104 · Mathematical Methods I — Linear Algebra & Operators

This unit builds the linear-algebraic machinery on which all of quantum mechanics, normal-mode analysis, and modern classical mechanics rest: vector spaces, inner products, linear operators, and above all the spectral theory of self-adjoint operators. Starting from abstract axioms and ending with the spectral theorem and its physical incarnations, it teaches the student to see eigenvalue problems, orthogonal decompositions, and unitary symmetries as one coherent structure.

Lectures

L01	What Linear Algebra Is For in Physics —
L02	Vector Spaces, Subspaces, and Linear Independence Invariance of Dimension of a Vector Space
L03	Bases and the Invariance of Dimension Invariance of Dimension of a Vector Space
L04	Linear Maps, Matrices, and Change of Basis Invariance of Dimension of a Vector Space
L05	Inner Products and Norms Cauchy–Schwarz and the Triangle Inequality
L06	Cauchy–Schwarz and Geometry of Hilbert Space Cauchy–Schwarz and the Triangle Inequality
L07	Orthogonality and Gram–Schmidt Gram–Schmidt Orthonormalization
L08	Dual Spaces and the Riesz Representation Riesz Representation in Finite Dimensions
L09	The Adjoint of an Operator Existence and Uniqueness of the Adjoint
L10	The Determinant: Volume and Multilinearity Determinant as the Unique Alternating Multilinear Form
L11	The Eigenvalue Problem and the Characteristic Polynomial Eigenvalues as Roots of the Characteristic Polynomial
L12	Diagonalization and When It Fails Diagonalizability and the Eigenbasis Criterion
L13	Hermitian Operators: Real Spectra, Orthogonal Modes Properties of Hermitian Operators
L14	The Spectral Theorem for Self-Adjoint Operators Spectral Theorem for Self-Adjoint Operators
L15	Projection Operators and the Resolution of the Identity Orthogonal Projections and Spectral Resolution
L16	Unitary Operators and Isometries Unitary Operators as Inner-Product Isometries
L17	Commuting Observables and Simultaneous Diagonalization Simultaneous Diagonalization of Commuting Operators
L18	Functions of Operators and the Spectral Calculus Spectral Theorem for Self-Adjoint Operators, Orthogonal Projections and Spectral Resolution
L19	The Matrix Exponential and Linear Flows Matrix Exponential and One-Parameter Flows
L20	Generators, Unitary Evolution, and Stone's Picture Matrix Exponential and One-Parameter Flows, Unitary Operators as Inner-Product Isometries
L21	The Rayleigh Quotient and Variational Eigenvalues Variational Characterization of Eigenvalues
L22	The Min-Max Principle and Eigenvalue Bounds Variational Characterization of Eigenvalues
L23	Coupled Oscillators and Normal Modes Normal Modes from the Generalized Eigenvalue Problem
L24	The Generalized Eigenvalue Problem and M-Orthogonality Normal Modes from the Generalized Eigenvalue Problem, Variational Characterization of Eigenvalues
L25	Quadratic Forms, Signature, and Principal Axes Spectral Theorem for Self-Adjoint Operators, Determinant as the Unique Alternating Multilinear Form
L26	The Singular Value Decomposition Singular Value Decomposition
L27	Least Squares, Pseudoinverses, and Low-Rank Approximation Singular Value Decomposition
L28	Tensor Products and Composite Systems Spectral Theorem for Self-Adjoint Operators, Simultaneous Diagonalization of Commuting Operators
L29	From Finite to Infinite Dimensions: A First Look Gram–Schmidt Orthonormalization, Spectral Theorem for Self-Adjoint Operators
L30	Synthesis: Operators as the Language of Physics Spectral Theorem for Self-Adjoint Operators, Properties of Hermitian Operators, Unitary Operators as Inner-Product Isometries

Derivations homed in this unit