Abstract Algebra
1. Algebraic Structure
1.3. Module-Like
1.3.2. Vector Space
See vector space for the definition.
1.3.2.1. Symplectic Vector Space
1.3.2.1.1. Definition
- Vector space over a field
equipped with a symplectic bilinear form
1.3.2.1.2. Standard Symplectic Space
with the symplectic form given by a nonsingular, skew-symmetric matrix, typically chosen to be:
1.3.2.1.3. Symplectic Map
- Linear map
between symplectic vector spaces , that the pullback preserves the symplectic form:
1.3.2.1.4. Symplectic Group
- Symplectic map
is called a linear symplectic transformation of . It preserves the symplectic form . - The set of symplectic transformations forms a
((66806d1f-9ca5-4f9d-b965-15197d286334)) called the symplectic
group, denoted
or . - Matrix form of symplectic transformations are symplectic matrices.
- The
1.3.2.1.5. Symplectic Complement
- For a linear subspace
, the symplectic complement of is:
1.3.2.1.6. Properties
- Unlike orthogonal complements,
need not be , distinguished four cases are: :- If and only if
restricts to a nondegenerate form on - Symplectic subspace with the restricted form is a symplectic vector space.
- If and only if
:- If and only if
restricts to 0 on - Any one-dimensional subspace is isotropic.
- If and only if
:- If and only if
descends to a nondegenerate form on the quotient space - Equivalently,
is coisotropic if and only if is isotropic. - Any codimension-one subspace is coisotropic
- If and only if
:- If and only if it is both isotropic and coisotropic
- In a finite-dimensional vector space, a Lagrangian subspace is
an isotropic one whose dimension is half that of
- Every isotropic subspace can be extended to a Lagrangian one.
1.3.2.1.7. Heisenberg Group
1.3.3. Algebra
2. Topological Structure
2.1. Topological Space
2.2. Metric Space
- Set with a notion of distance between its elements
2.2.1. Definition
- Ordered pair
where is a set, and is a metric on with:- Positivity:
- Symmetry:
- Triangle Inequality:
2.2.2. Properties
- It is also a topological space
2.2.3. Metrizable Space
- Topological Space that is homeomorphic to a metric space.
- There exists a metric
that induces the topology .
2.2.4. Closeness
- Arbitrarily near
2.2.4.1. Definition
- In a metric space
, a point is close or near to a set , if: where is the infimum. - Similarly, a set
is close to a set , if:
2.3. Normed Vector Space
- Normed Space
2.3.1. Definition
- Vector space
over on which a norm is defined, such that:- Non-Negativity:
- Positive Definiteness:
- Absolute Homogeneity:
- Triangle Inequality:
- Non-Negativity:
2.3.2. Properties
- It is also a metric space with the metric
induced by the norm: - If the norm satisfies the polarization identity then the inner product can be induced.
2.3.3. Polarization Identity
- Any formula that expresses the inner product in terms of the norm.
- Every inner product satisfies:
for the induced norm.
2.3.3.1. Theorem
- Norm satisfies the parallelogram law, if and only if,
there exists an inner product
such that for all . - The map between the norm and the inner product is bijective.
2.3.3.2. For Real Vector Space
- These forms are related by the Parallelogram law.
2.3.3.3. For Complex Vector Space
By stipulating the properties of the inner product:
if antilinear in the first argument,
if antilinear in the second argument.
2.3.4. Examples
2.3.4.1. Lp Space
Spaces, Lebesque Spaces
2.3.4.1.1. Definition
- Function spaces defined using a generalization of the
-norm--- -norm. - It is the space of measurable functions for which the
-norm is defined--- is Lebesgue integrable, modulo the equivalence relation .
- For a real number
:
2.3.4.1.2. Uniform Norm
- Sup Norm, Supremum Norm, Chebyshev Norm, Infinity Norm, Max Norm, Maximum Norm
- The uniform norm of a real- or complex-valued bounded functions
defined on a set is
2.3.4.2. Sobolev Space
2.3.4.2.1. Definition
- It is the subset of
- A normed vector space of functions equipped with a norm that is a combination of ((65ce2b90-4dde-4a95-964c-268f8ca744d1))s of a funtion and its derivatives up to a given order.
2.3.4.2.2. Norm
2.4. Banach Space
- Complete normed vector space
- The norm of Banach space is called complete norm, and the canonical metric is called the complete metric.
2.5. Inner Product Space
2.5.1. Definition
- Vector space
over the field —which can be either or —with an inner product satisfying:- Conjugate Symmetry:
- Linearity in the First Argument:
- Positive Definiteness:
- Conjugate Symmetry:
- It immediately follows from the definition that the inner product is antilinear in the second argument.
2.5.2. Properties
- It is also a normed vector space with the
norm induced by the inner product:
2.5.3. Vectors
2.5.3.1. Null Vector
2.5.3.2. Degenerate Vector
2.6. Hilbert Space
2.6.1. Definition
- inner product space whose induced metric space is complete.
2.6.1.1. Complete Metric Space
- If a series converges absolutely then the series converges.
- This is making sure that the limit exists.
2.6.2. Examples
2.6.2.1. L² Space
- Function space from measure space
to either or , equipped with the inner product: - This is the only Hilbert space among Lp spaces.
- Hilbert space - Wikipedia
2.7. Affine Space
- Informally, a vector space whose origin is forgotten by adding translations to the linear maps.
2.7.1. Definition
- A set
—points—with an associated vector space —isplacement vectors— and a transitive free action of the additive group of on the set , . - The action
satisfies:- Right Identity:
- Associativity:
- Free and Transitive action:
- Right Identity:
2.7.2. Affine Combination
- Linear combination in which the coefficients sums to 1.
- Affine combination of the points makes sense, which in turn can constitute a normalized barycentric coordinates.
2.7.3. Affine Structure
- Defined by the values of affine combination
2.8. Euclidean Space
- Basically,
with all the nice structures.
2.8.1. Definition
2.8.1.1. Euclidean Vector Space
- A Euclidean vector space is a finite-dimensional inner product space over the real numbers.
2.8.1.2. Euclidean Space
- Euclidean space is an affine space over the reals such that the associated vector space is a Euclidean vector space.
2.8.2. Euclidean Distance
- The Euclidean distance between two points in Euclidean space is the length of the line segment between them.
2.8.3. Properties
- Every Euclidean space of each dimension are all isomorphic to each
other. One of them is represented using Cartesian coordinate system as the real
-space such that the associated vector space is equipped with the standard dot product .
4. Order Structure
4.1. Preorder
- Quasiorder
,
4.1.1. Definition
- For elements that are comparable:
- Reflexive
- Transitive
- A set equipped with a preorder is called preordered set, proset.
4.1.2. Properties
- Antisymmetric preorder is a partial order, and symmetric preorder is an equivalence relation.
- Preorder induces an equivalence relation
with symmetry axiom: - Similarly it induces an strict partial order
with anti-symmetry axiom:
4.2. Partial Order
- Reflexive, Weak, or Non-Strict Partial Order
- The set equipped with partial order is called partially ordered set, poset.
- As the name suggests not every pair needs to be comparable.
4.2.1. Definition
4.2.2. Strict Partial Order
- Irreflexive, Strong, or Strict Partial Order
4.2.2.1. Definition
- Irreflexivity
- Asymmetry
- Transitivity
4.2.3. Reflexive Closure and Irreflexive Kernel
- Closure of the irreflexive relation is the union with the reflection.
- Kernel of the reflexive relation is the subtraction of the reflection.
4.3. Total Order
- Simple Order, Connex Order, Full Order
4.3.1. Definition
- Binary relation
on , that satisfies:- Reflexivity:
- Transitivity:
- Antisymmetry:
- Strongly Connected(Formerly, Total):
- Reflexivity:
- The set
is called totally ordered set, simply ordered set, linearly ordered set, loset. - Chain is generally the totally ordered subset of a partially ordered set.
4.3.2. Strict Total Order
- Binary relation
on
4.3.3. Definition
- Irreflexivity:
- Asymmetry:
- Transitivity:
- Connected:
4.4. Directed Set
4.4.1. Definition
4.4.2. Upward and Downward Directed Sets
- Upward directed set requires the existance of the common upper bound, and Downward directed set requires that of the common lower bound.
- excalidraw:directed_sets.excalidraw
4.5. Hasse Diagram
- Mathematical diagram used to represent a partially ordered set.
- Larger objects on top, and smaller objects on the bottom, with the containment relationships indicated with lines.
5. Composite Structure
5.0.1. Vector Bundle
- Set of vector spaces.
- e.g. Tangent bundle
5.0.2. Fiber Bundle
- Topological Concept
- See fiber bundle
6. Operations
6.1. Element-wise
6.1.1. Formal Product
- Form of a product with no additional structure.
6.1.2. Tensor Product
- The tensor product
is then defined as , where is the coset.
6.1.3. Wedge Product
- See exterior product
6.1.4. Dot Product
6.1.5. Pseudo-Inner Product
6.1.5.1. Definition
- Bilinear (or sesquilinear)
- Symmetric
- Non-Degenerate
- See degeneracy
- Weakening of the positive-definiteness.
6.1.7. Geometric Product
6.2. Structure-wise
6.2.1. Formal Product
- Given two vector spaces
and , forms a vector space of dimension .
6.2.2. Tensor Product
: tensor producted with itself times.- Tensor product of two vector spaces
is the quotient vector space with respect to the subspace :
6.2.2.1. Properties
then
6.2.3. Direct Product
- This is what is meant by
.
6.2.3.1. Properties
then
6.2.4. Direct Sum
- The two sets become just one set.
- Direct Product with inclusions defined.
6.2.4.1. Properties
- It is the coproduct
7. Quotient
- It is the general idea of forming a space of equivalence classes generated by dividing the collection with a equivalence relation.
7.1. Quotient Group
- See quotient group
7.2. Quotient Ring
- See quotient ring
7.3. Quotient Space
- Quotient of a Vector Space
- The quotient space
consists of equivalence classes defined by the equivalence relation : where is a linear subspace of .- The equivalence class (or the coset) of
is denoted: - The scalar multiplication and addition are defined as:
- The equivalence class (or the coset) of
7.4. Quotient Space
- Quotient Space of a Topological Space
- Topological of equivalence classes generated by a equivalence relation.
8. Form
A function from a vector space to the underlying field. It may have a corresponding representation in the matrix form.
8.1. Bilinear Form
8.1.1. Definition
A bilinear form
8.1.1.1. Symmetric
8.1.1.2. Skew-Symmetric
8.1.1.3. Altenating
8.2. Sesquilinear Form
- sesqui-, one and a half.
8.2.1. Definition
- One of its arguments is semilinear: antilinear, and others.
8.2.2. Antilinearity
, where denotes the complex conjugate.- I just want to call it semilinearity, but there so much other things that are called semilinear.
8.3. Symplectic Form
8.3.1. Symplectic Bilinear Form
- Mapping
that has following properties:- Bilinear
- Alternating:
- Non-degenerate:
- If the underlying field has characteristic
not 2, alternation is equivalent to skew-symmetry:
. - If the characteristic is 2, alternation implies skew-symmetry but not in the other direction.
8.3.2. Example
- 2-Form
8.4. Quadratic Form
8.4.1. Definition
- Homogeneous polynomial of degree two. ("form" is another name for a homogeneous polynomial1)
- Quadratic form on a vector space
over a field is such that .
8.4.2. Properties
- Every quadratic form has an associated symmetric matrix:
- That is every quadratic form can be orthogonally diagonalized using
the spectral decomposition:
9. Isomorphism
9.1. Definition
- Structure-preserving mapping between two structures that can be reversed by an inverse mapping.
9.2. Canonical Isomorphism
- The most trivial isomorphism between two structures.
9.3. Special Isomorphisms
- Isometry: Metric Spaces
- Homeomorphism: Topological Spaces
- Diffeomorphism: Differentiable Manifolds
- Symplectomorphism: Symplectic Manifolds
- Permutation: Automorphism of a Set
- Transformation: Geometric Objects
9.4. Isomorphism Theorems for Groups
9.4.1. First
- For a group homomorphism
,
9.4.2. Second
- Let
and be a two normal subgroups of a group ,
9.4.3. Third
- For a subgroup
, and a normal subgroup ,
9.5. Isomorphism Theorems for Rings
9.5.1. First
- For a ring homomorphism
, there exists a unique isomorphism such that .
10. Tensor
10.1. Intuitions
- Tensor is multidimensional array
- Tensor is multilinear map
- Tensor is tensor product of vectors and covectors
- Tensor (from physics) is an object that transforms like a tensor
10.2. Multilinear Map
- Multilinear map from
to , corresponds to the (p,q)-tensor.
10.3. Definition
- Given a finite set of vector spaces
over a common field , the element of their tensor product is a tensor :
10.3.1. On a Vector Space
- Tensor
of type on a vector space is defined as:
10.4. Type
- Order, Rank, Valence, Degree
- Pair of orders of contravariance and covariance.
. contravariant components (vectors), covariant components (covectors).
10.5. Order
- Degree, Rank
- The dimension of a tensor
.
10.6. Rank
- The minimum number of simple tensors that sum to the tensor.
- Often used to mean the order of a tensor.
10.7. Universal Property
- From the universal characterization of the tensor product, the
space of (p,q)-tensors admits a natural isomorphism:
when is finite dimensional. is denoting the space of n-linear maps from to .
10.8. (1, 1)-Tensor
10.8.1. Geometric Interpretation
- Visualization of tensors - part 1 - YouTube
- Symmetric matrix has a transformation into a diagonal matrix.
- Anti-symmetric matrix project a vector to a plane and rotate it
by
.
- Visualization of tensors - part 2A - YouTube
- Magnetic field can be represented by a anti-symmetric tensor field.
- By adding electric field in the time column we have the electromagnetic tensor.
10.8.2. Examples
10.9. Tensor Contraction
- It is the generalization of trace
- For a tensor of type
, contraction is a linear map from type -tensor to type -tensor defined by the canonical pairing of th vector space and th dual vector space: - The application of tensor is also a contraction:
- For a tensor of type
10.10. Tensor Field
10.10.1. Definition
- A tensor field
of type on a manifold is: - where
is a vector bundle on .
11. Tensor Algebra
11.1. Definition
- Tensor algebra
of a vector space over af field is:
11.1.1. Tensor Power
is called the th tensor power of with , and the multiplication is the linear extension of canonical isomorphism:- Note that
of is not a tensor product.
- Note that
11.2. Properties
- It is the functor form the category of vector spaces
to the category of algebras
12. Exterior Algebra
12.1. Definiton
- The quotient algebra of the tensor algebra
with the ideal :
12.2. Exterior Power
th exterior power of is the vector subspace of the exterior algebra with grade is
12.3. Exterior Product
12.3.1. Of Tensors
- For
, - where
denotes the equivalence classes with respect to .
12.3.2. Of Differential Forms
12.4. Grassmann Algebra
can be thought of as the space of all forms on ,- equipped with addition
, the direct sum on elements, and the scalar multiplication , and the bilinear map that is a linear continuation of wedge product: .
13. Exact Sequence
13.1. Definition
- Sequence of morphisms between objects
- such that it is exact at each
, that is, .
13.2. Properties
- For the sequence of group homomorphisms,
is exact, if and only if is a monomorphism. is exact, if and only if is a epimorphism.
13.3. Short Exact Sequence
- Exact sequence of the form
13.3.1. Properties
- Intuitively:
is a subobject of and is the correspoding factor object
13.3.2. Split Exact Sequence
- A short exact sequence is called split if there exists a
homomorphism
such that . - If the objects are abelian groups,
.
13.4. Splitting Lemma
- In any abelian category, for a short exact sequence
- the following statements are equivalent:
- Left Split:
- Right Split:
- Direct Sum: logseq.order-list-type:: number
- Left Split:
14. References
- Metrizable space - Wikipedia
- Metric space - Wikipedia
- Sobolev space - Wikipedia
- Normed vector space - Wikipedia
- Fourier Transform 5 | Integrable Functions - YouTube
- Lp space - Wikipedia
- Uniform norm - Wikipedia
- Inner product space - Wikipedia
- Affine space - Wikipedia
- Affine combination - Wikipedia
- Euclidean space - Wikipedia
- Lebesgue measure - Wikipedia
- Hilbert space - Wikipedia
- Tensor (intrinsic definition) - Wikipedia
- Tensor field - Wikipedia
- Tensor contraction - Wikipedia
- Measure (mathematics) - Wikipedia
- σ-algebra - Wikipedia
- Radon–Nikodym theorem - Wikipedia
- Preorder - Wikipedia
- Partially ordered set - Wikipedia
- Total order - Wikipedia
- Directed set - Wikipedia
- Quotient space (linear algebra) - Wikipedia
- Quotient space (topology) - Wikipedia
- Multilinear map - Wikipedia
- Isomorphism - Wikipedia
- Symplectic vector space - Wikipedia
- Tensor algebra - Wikipedia
- Exterior algebra - Wikipedia
- Exact sequence - Wikipedia
- Split exact sequence - Wikipedia
- Splitting lemma - Wikipedia
- Splitting Lemma in homological algebra, not to be confused with the splitting lemma in singularity theory.