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Mathematics Notes
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Linear Algebra
Linear Algebra
Ch 1. Vector Spaces
Ch 1. Vector Spaces
2. Vector Spaces
3. Subspaces
4. Linear Combinations and Systems of Linear Equations
5. Linear Dependence and Linear Independence
6. Bases and Dimension
7. Maximal Linearly Independent Subsets
Ch 2. Linear Transformations and Matrices
Ch 2. Linear Transformations and Matrices
8. Linear Transformations, Null spaces, and Ranges
9. The Matrix Representation of a Linear Transformation
10. Composition of Linear Transformations and Matrix Multiplication
11. Invertibility and Isomorphisms
12. The Change of Coordinate Matrix
Ch 3. Elementary Matrix Operations and Systems of Linear Equations
Ch 3. Elementary Matrix Operations and Systems of Linear Equations
15. Elementary Matrix Operations and Elementary Matrices
16. The Rank of a Matrix and Matrix Inverses
17. Systems of Linear Equations - Theoretical Aspects
18. Systems of Linear Equations - Computational Aspects
Ch 4. Determinants
Ch 4. Determinants
19. Determinants of Order 2
20. Determinants of Order n
21. Properties of Determinants
Ch 5. Diagonalization
Ch 5. Diagonalization
23. Eigenvalues and Eigenvectors
24. Diagonalizability
26. Invariant Subspaces and the Cayley-Hamliton Theorem
Ch 6. Inner Product Spaces
Ch 6. Inner Product Spaces
27. Inner Products and Norms
28. The Gram-Schmidt Orthogonalization Process and Orthogonal Complements
29. The Adjoint of a Linear Operator
30. Normal and Self-Adjoint Operators
31. Unitary and Orthogonal Operators and Their Matrices
32. Orthogonal Projections and the Spectral Theorem
33. The Singular Value Decomposition and the PseudoInverse
Abstract Algebra
Abstract Algebra
0. Sets and Relations
Ch 1. Groups and SubGroups
Ch 1. Groups and SubGroups
1. Introduction and Examples
2. Binary Operations
3. Isomorphic Binary Structures
4. Groups
5. Subgroups
6. Cyclic Groups
7. Generating Sets and Cayley Digraphs
Ch 2. Permutations, Cosets, and Direct Products
Ch 2. Permutations, Cosets, and Direct Products
8. Groups of Permutations
9. Orbits, Cycles, and the Alternating Groups
10. Cosets and the Theorem of Lagrange
11. Direct Products and Finitely Generated Abelian Groups
12. Plane Isometries
Ch 3. Homomorphisms and Factor Groups
Ch 3. Homomorphisms and Factor Groups
13. Homomorphisms
14. Factor Groups
15. Factor-Group Computations and Simple Groups
16. Group Action on a Set
17. Applications of G-Sets to Counting
Ch 4. Rings and Fields
Ch 4. Rings and Fields
18. Rings and Fields
19. Integral Domains
20. Fermat's and Euler's Theorem
21. The Field of Quotients of an Integral Domain
22. Rings of Polynomials
23. Factorization of Polynomials over a Field
24. Noncommutative Examples
25. Ordered Rings and Fields
Topology
Topology
Ch 1. Set Theory and Logic
Ch 1. Set Theory and Logic
1. Fundamental Concepts
2. Functions
3. Relations
4. The Integers and the Real Numbers
5. Cartesian Products
6. Finite Sets
Ch 2. Topological Spaces and Continuous Functions
Ch 2. Topological Spaces and Continuous Functions
12. Topological Spaces
13. Basis for a Topology
14. The Order Topology
15. The Product Topology on X ⨯ Y
16. The Subspace Topology
17. Closed Sets and Limit Points
18. Continuous Functions
19. The Product Topology
20. The Metric Topology
21. The Metric Topology (continued)
22. The Quotient Topology
Ch 3. Connectedness and Compactness
Ch 3. Connectedness and Compactness
23. Connected Spaces
24. Connected Subspaces of the Real Line
25. Components and Local Connectedness
26. Compact Spaces
27. Compact Subspaces of the Real Line
28. Limit Point Compactness
29. Local Compactness
목차
Topology (James Munkres)
Topology
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Topology (James Munkres)
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