Math Study
1. Set Theory
Set Theory 1
- set
- power_set
- function
- inverse
- enjoying_math
Set Theory 2
- injective
- 1-1
- surjective
- onto
- bijective
- inverse
- enjoying_math
Set Theory 3
- relation
- equivalence_relation
- partition
- enjoying_math
2. Mathematical Analysis
Completeness Axiom
- least_upper_bound_property
- boundedness
- supremum
- enjoying_math
Denseness of Rational Numbers
- mathematical_induction
- well-ordering_principle
- archimedean_property
- enjoying_math
Convergence of Sequences
- sequence
- absolute_value
- boundedness
- convergence
- enjoying_math
Limit Theorems
- limit_theorem
- enjoying_math
ε-δ and the Convergence of a Sequence
- limit_of_function
- continuity_of_function
- sandwich_theorem
- enjoying_math
MCT & NIP
- monotone_convergence_theorem
- MCT
- nested_interval_property
- NIP
liminf & limsup
- limsup
- liminf
- Bolzano-Weierstrass-Property
3. Topology
Definition of Topological Space
- lumped_element
- topology
- continuous_function
- cofinite_topology
- topological_space
- discrete_topological_space
- indiscrete_topological_space
- fineness
- coarseness
- enjoying_math
Mathematical Analysis & Metric Topology
- metric_topology
- epsilon-neighborhood
- openness
- continuity_of_function
- convergence_of_sequence
- Hausdorff-axiom
- T2-axiom
- enjoying_math
4. Algebra
Concepts in Algebra
- binary_operation
- semi-group
- identity
- monoid
- group
- commutative_group
- abelian_group
- Lie-bracket
- ring
- field
- module
- vector_space
- algebra
5. Linear Algebra
Vector Space & Zorn's Lemma
- vector_space
- span
- linear_independence
- basis
- partially_ordered_set
- POSET
- totally_ordered_set
- maximal
- minimal
- chain
- zorn's_lemma
Existence of Basis & Dimensionality
- zero_dimensional_vector_space
- cardinality
- dimension
Dimensionality & Various Vector Spaces
- coordinate
- inner_product
- linear_transformation
- isomorphism
- equivalence_relation
Matrix Representation of the Linear Transformation
- matrix_representation
- linear_transformation
Determinant
- determinant
- elementary_row_operations
Coordinate Change
- coordinate_change_matrix
- $\left([T]_\beta\right)$
- similarity
- $(\backsim)$
Eigenvectors & Diagonalization 1
- eigenvector
- eigenvalue
- diagonalizable
- characteristic_polynomial_of_matrix
- $\left(\varphi_A(t)\right)$
Eigenvectors & Diagonalization 2
- linear_operator_determinant
- linear_operator_trace
- linear_operator_characteristic_polynomial
Diagonalization
- eigenspace
- $(E_{\lambda})$
- dimension_formula
- rank_nullity_theorem
- sum_of_sets
- direct_sum_of_sets
- $(\oplus)$
6. Abstract Algebra
Abstraction of Linear Combination
- smallest_subsequence
- subspace
- span
- subgroup
Homomorphism
- group-homomorphism
- monomorphism
- epimorphism
- isomorphism
- linear_transformation_as_module-homomorphism
- kernel
- image
- nullity
- rank
- dimension
Cyclic Group
- cyclic_group
- quotient
- $$\mathbb{Z}/n\mathbb{Z}$$
- division_algorithm
Inverse of Lagrange's Theorem
- order
- euler-phi-function
Coset & Lagrange's Theorem
- coset
- left_coset
- right_coset
- lagrange_theorem
Group Action
- group_action
- conjugate_action
- permutation_representation
- Cayley_Theorem
- orbit
Normal Subgroup
- normal_subgroup
- ($\unlhd$)
1-1 Correspondence of Normal Subgroups
- simple_group
- maximal_normal_subgroup
Isomorphism Theorems
- isomorphism_theorems
Symmetric Group
- symmetric_group
- permutation
- cyclic_notation
- transposition
- conservation_of_parity
- even
- signum_funcion
- alternating_subgroup
- determinant
Direct Product & Direct Sum
- internal_direct_product
- external_direct_product
- direct_sum
Completeness Axiom
2. Mathematical Analysis- least_upper_bound_property
- boundedness
- supremum
- enjoying_math
Denseness of Rational Numbers
2. Mathematical Analysis- mathematical_induction
- well-ordering_principle
- archimedean_property
- enjoying_math
Convergence of Sequences
2. Mathematical Analysis- sequence
- absolute_value
- boundedness
- convergence
- enjoying_math
Definition of Topological Space
3. Topology- lumped_element
- topology
- continuous_function
- cofinite_topology
- topological_space
- discrete_topological_space
- indiscrete_topological_space
- fineness
- coarseness
- enjoying_math
Mathematical Analysis & Metric Topology
3. Topology- metric_topology
- epsilon-neighborhood
- openness
- continuity_of_function
- convergence_of_sequence
- Hausdorff-axiom
- T2-axiom
- enjoying_math
ε-δ and the Convergence of a Sequence
2. Mathematical Analysis- limit_of_function
- continuity_of_function
- sandwich_theorem
- enjoying_math
Concepts in Algebra
4. Algebra- binary_operation
- semi-group
- identity
- monoid
- group
- commutative_group
- abelian_group
- Lie-bracket
- ring
- field
- module
- vector_space
- algebra
Vector Space & Zorn's Lemma
5. Linear Algebra- vector_space
- span
- linear_independence
- basis
- partially_ordered_set
- POSET
- totally_ordered_set
- maximal
- minimal
- chain
- zorn's_lemma
Existence of Basis & Dimensionality
5. Linear Algebra- zero_dimensional_vector_space
- cardinality
- dimension
Dimensionality & Various Vector Spaces
5. Linear Algebra- coordinate
- inner_product
- linear_transformation
- isomorphism
- equivalence_relation
Matrix Representation of the Linear Transformation
5. Linear Algebra- matrix_representation
- linear_transformation
Homomorphism
6. Abstract Algebra- group-homomorphism
- monomorphism
- epimorphism
- isomorphism
- linear_transformation_as_module-homomorphism
- kernel
- image
- nullity
- rank
- dimension
Cyclic Group
6. Abstract Algebra- cyclic_group
- quotient
- $$\mathbb{Z}/n\mathbb{Z}$$
- division_algorithm
Group Action
6. Abstract Algebra- group_action
- conjugate_action
- permutation_representation
- Cayley_Theorem
- orbit
Symmetric Group
6. Abstract Algebra- symmetric_group
- permutation
- cyclic_notation
- transposition
- conservation_of_parity
- even
- signum_funcion
- alternating_subgroup
- determinant
Direct Product & Direct Sum
6. Abstract Algebra- internal_direct_product
- external_direct_product
- direct_sum
Coordinate Change
5. Linear Algebra- coordinate_change_matrix
- $\left([T]_\beta\right)$
- similarity
- $(\backsim)$
Eigenvectors & Diagonalization 1
5. Linear Algebra- eigenvector
- eigenvalue
- diagonalizable
- characteristic_polynomial_of_matrix
- $\left(\varphi_A(t)\right)$
Eigenvectors & Diagonalization 2
5. Linear Algebra- linear_operator_determinant
- linear_operator_trace
- linear_operator_characteristic_polynomial
Diagonalization
5. Linear Algebra- eigenspace
- $(E_{\lambda})$
- dimension_formula
- rank_nullity_theorem
- sum_of_sets
- direct_sum_of_sets
- $(\oplus)$