Notes for MAT3040, MAT3006, MAT4002
(Term 2, 2018-19)
Lecture Notes for MAT3040
Errata Correction: the Quotient Space should be written as V/W instead of V\W.
- Lecture 1:
Introduction, Vector Spaces
-
Lecture 2: Spanning Set, Linear Independence, Basis
- Lecture 3:
Basis, Dimension, Operations on a vector space
- Lecture 4:
Direct Sum, Linear Transformation,
- Lecture 5:
Isomorphism, Change of Basis, Matrix Representation,
- Lecture 6:
Change of Basis,
- Lecture 7:
Quotient Spaces, First Isomorphism Theorem
- Lecture 8:
Quotient Spaces, Dual Space
- Lecture 9:
Dual Space, Annihilators
- Lecture 10:
Adjoint Map, Annihilator, Dual of Quotient Spaces
- Lecture 11:
Evil Quiz, Preliminaries on polynomials
- Lecture 12:
Preliminaries on polynomials, Eigenvalues
- Lecture 13:
Minimal Polynomial, Minimal Polynomial of a vector
- Lecture 14:
Cayley-Hamiton Theorem
- Lecture 15:
Cayley-Hamiton Theorem, Primary Decomposition Theorem
- Lecture 16:
Evil Mid-term
- Lecture 17:
Remarks on Primary Decomposition Theorem, Spectral Decomposition Theorem
- Lecture 18:
Jordan Normal Form
- Lecture 19:
Inner Product Space, Riesz Representation
- Lecture 20:
Orthogonal Complement, (Hermitian) Adjoint Map
- Lecture 21:
Remarks on Adjoint Map
- Lecture 22:
Unitary Operator, Normal Operator
- Lecture 23:
Remarks on Normal Operator
- Lecture 24:
Introduction to Tensor Product
- Lecture 25:
Basis for tensor product, universal property
- Lecture 26:
Tensor Product for Linear Transformations
- Lecture 27:
Multi-linear Tensor Product, Exterior Power, Final Exam Ends Here
- Collection of Notes for MAT3040
This is the collection of notes during this semester. Thanks Prof. Daniel and other students for giving some suggestions, which help improving the quality of this note!
Lecture Notes for MAT3006
I want to thank Mr.Yige Hong for his detailed and valuable refinement on week11 notes, which significantly improved the quality of the notes.
Errata Correction: updated later.
- Lecture 1:
Overview, Metric Spaces
-
Lecture 2: Convergence, Continuity, Open sets, Closed sets
- Lecture 3:
Open sets, Closed sets, Boundary, Closure, Interior
- Lecture 4:
Compactness, Completeness,
- Lecture 5:
Completeness, Contraction Mapping Theorem,
- Lecture 6:
Contraction, Picard Lindelof Theorem,
- Lecture 7:
System of ODEs, Stone-Weierstrass Theorem
- Lecture 8:
Stone-Weierstrass Theorem
- Lecture 9:
Fourier Analysis, Baire Category Theorem
- Lecture 10:
Evil Quiz, Baire Category Theorem, Compact Subsets on C[a,b]
- Lecture 11:
Arzela-Ascoli Theorem
- Lecture 12:
Introduction to Measure Theory (to be updated)
- Lecture 13:
Outer Measure, Lebesgue Measure
- Lecture 14:
Evil Mid-term
- Lecture 15:
Remarks on Outer Measure, Lebesgue Measurable
- Lecture 16:
Remarks on Lebesgue Measurable, Probability Theory
- Lecture 17:
Measurable Function
- Lecture 18:
Remarks on Measurable Function, Lebesgue Integral
- Lecture 19:
Remarks on Markov Inequality, Properties of Lebesgue Integration
- Lecture 20:
Fatou's Lemma, Monotone Convergence Theorem
- Lecture 21:
Generalization of MCT for almost everywhere case
- Lecture 22:
Properties of Lebesgue Integration without non-negativity
- Lecture 23:
Remarks on MCT, DCT
- Lecture 24:
Riemann integrability & Lebesgue integrability, Continous parameter DCT
- Lecture 25:
Interchange of integral and partial derivative, Double Integral
- Lecture 26:
Introduction to Fubini's and Tonell's Theorem
- Lecture 27:
Fubini's and Tonell's Theorem
- Lecture 28:
Applications for Fubini's and Tonell's Theorem, Final Exam Ends Here
- Collection of Notes for MAT3006
This is the collection of notes during this semester. Thanks Prof. Daniel and other students for giving some suggestions, which help improving the quality of this note!
Lecture Notes for MAT4002
I want to thank Mr.Yige Hong for his detailed and valuable refinement on week11 notes, which significantly improved the quality of the notes.
Errata Correction: updated later.
- Lecture 1:
Introduction, Metric Spaces
-
Lecture 2: Forget about metric, Topological Spaces
- Lecture 3:
Convergence, Interior, Closure, Boundary
- Lecture 4:
Closure, Mapping, Subspace Topology, Basis,
- Lecture 5:
Basis, Homeomorphism, Product space,
- Lecture 6:
Product Space, Properties of Topological Spaces,
- Lecture 7:
Hausdorffness, Connectedness
- Lecture 8:
Topogist's Bomb, Completeness
- Lecture 9:
Evil Quiz, Continuous Functions on Compact space
- Lecture 10:
Compactness, Quotient Spaces
- Lecture 11:
Quotient Topology, Homeomorphism of torus and sphere
- Lecture 12:
Quotient Mapping
- Lecture 13:
Simplicial Complex
- Lecture 14:
Simplicial sub-complex
- Lecture 15:
Evil Mid-term
- Lecture 16:
Remarks on simplicial subcomplex, Homotopy
- Lecture 17:
Remarks on Homotopy
- Lecture 18:
Simplicial Approximation Theorem
- Lecture 19:
Simplicial Approximation Theorem
- Lecture 20:
Free Group, Group with Relations
- Lecture 21:
Group Presentations, Fundamental Group
- Lecture 22:
The Fundamental Group
- Lecture 23:
Remarks on the Fundamental Group
- Lecture 24:
Groups & Simplicial Complices
- Lecture 25:
Isomorphism of Fundamental Group makes life easier
- Lecture 26:
More Applications on the Isomorphism of Fundamental Group
- Lecture 27:
Fundamental group of a Graph
- Lecture 28:
The Selfert-Van Kampen Theorem: Computing Fundamental Group of a Graph
- Collection of Notes for MAT4002
This is the collection of notes during this semester. Thanks Prof. Daniel and other students for giving some suggestions, which help improving the quality of this note!