Abstract Algebra

• Group without the inverse elements.

See vector space for the definition.

• Ordered pair $(M,d)$ where $M$ is a set, and $d:M\times M\to \mathbb{R}$ is a metric on $M$ with:
  • $d(x,x)=0$
  • Positivity: $x\ne y⟹d(x,y)>0$
  • Symmetry: $d(x,y)=d(y,x)$
  • Triangle Inequality: $d(x,z)\le d(x,y)+d(y,z)$

• It is also a topological space

• Topological Space that is homeomorphic to a metric space.
• There exists a metric $d$ that induces the topology $\tau$.

• Arbitrarily near

• Vector space $V$ over $K$ on which a norm $\Vert \cdot \Vert$ is defined, such that:
  • Non-Negativity: $\mathrm{\forall }x\in V,\Vert x\Vert \ge 0$
  • Positive Definiteness: $\mathrm{\forall }x\in V,(\Vert x\Vert =0⟺x=\mathbf{0})$
  • Absolute Homogeneity: $\mathrm{\forall }\lambda \in K,\mathrm{\forall }x\in V,\Vert \lambda x\Vert =|\lambda |\Vert x\Vert$
  • Triangle Inequality: $\mathrm{\forall }x,y\in V,\Vert x+y\Vert \le \Vert x\Vert +\Vert y\Vert$

• It is also a metric space with the metric $d$ induced by the norm: $$d(x,y)=\Vert y-x\Vert .$$
• If the norm satisfies the polarization identity then the inner product can be induced.

• Any formula that expresses the inner product in terms of the norm.
• Every inner product satisfies: $$\Vert x+y{\Vert }^2=\Vert x{\Vert }^2+\Vert y{\Vert }^2+2\mathfrak{R}⟨x,y⟩$$ for the induced norm.

• Vector space $V$ over the field $F$—which can be either $\mathbb{R}$ or $\mathbb{C}$—with an inner product $⟨\cdot ,\cdot ⟩:V\times V\to F$ satisfying:
  • Conjugate Symmetry: $⟨x,y⟩=\overline{⟨y,x⟩}$
  • Linearity in the First Argument: $⟨ax+by,z⟩=a⟨x,z⟩+b⟨y,z⟩$
  • Positive Definiteness: $x\ne 0⟹⟨x,x⟩>0$
• It immediately follows from the definition that the inner product is antilinear in the second argument.

• It is also a normed vector space with the norm induced by the inner product: $$\Vert x\Vert =\sqrt{⟨x,x⟩}.$$

• Null Vectors vs. Degenerate Vectors - YouTube

• inner product space whose induced metric space is complete.

• A set $A$—points—with an associated vector space $\overrightarrow{A}$—isplacement vectors— and a transitive free action of the additive group of $\overrightarrow{A}$ on the set $A$, $+$.
• The action $+:A\times \overrightarrow{A}\to A$ satisfies:
  • Right Identity: $$\mathrm{\forall }a\in A,a+0=a.$$
  • Associativity: $$\mathrm{\forall }v,w\in \overrightarrow{A},\mathrm{\forall }a\in A,(a+v)+w=a+(v+w).$$
  • Free and Transitive action: $$\mathrm{\forall }a\in A,\text{the mapping }\overrightarrow{A}\to A:v\mapsto a+v\text{ is a bijection}.$$

• 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.

• Defined by the values of affine combination

• The Euclidean distance between two points in Euclidean space is the length of the line segment between them.

• Every Euclidean space of each dimension are all isomorphic to each other. One of them is represented using Cartesian coordinate system as the real $n$-space ${\mathbb{R}}^n$ such that the associated vector space is equipped with the standard dot product .

• For elements that are comparable:
  • Reflexive
  • Transitive
• A set equipped with a preorder is called preordered set, proset.

• Antisymmetric preorder is a partial order, and symmetric preorder is an equivalence relation.
• Preorder induces an equivalence relation $\sim$ with symmetry axiom: $$a\sim b⟺a\lesssim b\wedge a\gtrsim b.$$
• Similarly it induces an strict partial order $<$ with anti-symmetry axiom: $$a<b⟺a\lesssim b\wedge \mathrm{\neg }(a\gtrsim b).$$

• For the pairs that the partial order is defined:
  • Reflexivity
  • Antisymmetry: $a\le b\wedge b\le a⟹a=b.$
    • Therefore, the order relation on $\mathbb{C}$ is not partial order, rather, preorder.
  • Transitivity
• It is an antisymmetric preorder.

• Irreflexive, Strong, or Strict Partial Order

• Closure of the irreflexive relation is the union with the reflection.
• Kernel of the reflexive relation is the subtraction of the reflection.

• Binary relation $\le$ on $X$, that satisfies:
  • Reflexivity: $a\le a.$
  • Transitivity: $a\le b\wedge b\le c⟹a\le c.$
  • Antisymmetry: $a\le b\wedge b\le a⟹a=b.$
  • Strongly Connected(Formerly, Total): $a\le b\vee b\le a.$
• The set $(X,\le )$ is called totally ordered set, simply ordered set, linearly ordered set, loset.
• Chain is generally the totally ordered subset of a partially ordered set.

• Binary relation $<$ on $X$

• Irreflexivity: $\mathrm{\neg }(a<a).$
• Asymmetry: $a<b⟹\mathrm{\neg }(b<a).$
• Transitivity: $a<b\wedge b<c⟹a<c.$
• Connected: $a\ne b⟹a<b\vee b<a.$

• Set with a preorder $\lesssim$, where every pair of elements has an upper bound.
• The preorder of a directed set is called a direction.

• 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

• $\ast$
• Form of a product with no additional structure.

• $\otimes$
• The tensor product $v\otimes w$ is then defined as $v\ast w+I$, where $I$ is the coset.

• $\wedge$
• See exterior product

• See geometric product

• $\ast$
• Given two vector spaces $V$ and $W$, $V\ast W={\mathrm{span}}_{\mathbb{R}}\{a\ast b\mid a\in V,b\in W\}$ forms a vector space of dimension $|V|\cdot |W|$.

• $\otimes$
• $V^{\otimes n}$: $V$ tensor producted with itself $n$ times.
• Tensor product of two vector spaces $V\oplus W$ is the quotient vector space $V\ast W/I$ with respect to the subspace $I$: $$I={\mathrm{span}}_{\mathbb{R}}\{\overline{)\begin{array}{l}(cv)\ast w-c(v\ast w),\\ v\ast (cw)-c(v\ast w),\\ (v_1+v_2)\ast w-(v_1\ast w+v_2\ast w),\\ v\ast (w_1+w_2)-(v\ast w_1+v\ast w_2)\end{array}}c\in \mathbb{R},v\in V,w\in W\}$$

• $\times$
• This is what is meant by ${\mathbb{R}}^n$.

• $\oplus$
• The two sets become just one set.
• Direct Product with inclusions defined.

A bilinear form $B$ is a bilinear map on a vector space $V$ over a field $K$: $$B:V\times V\to K$$ such that it's linear in both arguments.

• One of its arguments is semilinear: antilinear, and others.

• $\phi (ax)=a^{\ast }\phi (x)$ , where ${}^{\ast }$ denotes the complex conjugate.
• I just want to call it semilinearity, but there so much other things that are called semilinear.

• Mapping $\omega :V\times V\to F$ that has following properties:
  • Bilinear
  • Alternating: $\mathrm{\forall }v\in V,\omega (v,v)=0$
  • Non-degenerate: $(\mathrm{\forall }v\in V,\omega (v,u)=0)⟹u=0$
• If the underlying field has characteristic not 2, alternation is equivalent to skew-symmetry: $\omega (v,u)=-\omega (u,v)$.
• If the characteristic is 2, alternation implies skew-symmetry but not in the other direction.

• 2-Form

• Homogeneous polynomial of degree two. ("form" is another name for a homogeneous polynomial¹)
• Quadratic form on a vector space $V$ over a field $K$ is $$q:V\to K$$ such that $q(a\mathbf{v})=a^{2}q(\mathbf{v})$.

• Every quadratic form has an associated symmetric matrix: $$q(\mathbf{x})={\mathbf{x}}^{\mathrm{T}}A\mathbf{x}.$$
• That is every quadratic form can be orthogonally diagonalized using the spectral decomposition: $${\mathbf{x}}^{\mathrm{T}}\mathbf{A}\mathbf{x}={\mathbf{x}}^{\mathrm{T}}Q\mathrm{\Lambda }{Q}^{\mathrm{T}}\mathbf{x}={\tilde{\mathbf{x}}}^{\mathrm{T}}\mathrm{\Lambda }\tilde{\mathbf{x}}$$

• For a group homomorphism $\phi :G\to H$,
• $$G/\ker \phi \cong H$$

• Let $H$ and $N$ be a two normal subgroups of a group $G$,
• $$G/H\cong (G/N)/(H/N)$$

• For a subgroup $H\le G$, and a normal subgroup $N◃G$,
• $$H/(H\cap N)\cong (HN)/N$$

• For a ring homomorphism $\phi :R\to S$, there exists a unique isomorphism $\psi :R/\ker \phi \to \mathrm{im}\phi$ such that $\psi (r+\ker \phi )=\phi (r)$.

• Tensor $T$ of type $(p,q)$ on a vector space $V$ is defined as: $$T\in T_q^p(V):=V^{\otimes p}\otimes (V^{\ast }{)}^{\otimes q}.$$

• 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 $\pi /2$.
• 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.

• Cauchy stress tensor

• A tensor field $T$ of type $(p,q)$ on a manifold $M$ is: $$T\in \mathrm{\Gamma }(M,V^{\otimes p}\otimes (V^{\ast }{)}^{\otimes q}),$$
• where $V$ is a vector bundle on $M$.

• $T^{k}V$ is called the $k$th tensor power of $V$ with $T^{0}V:=K$, and the multiplication $\otimes$ is the linear extension of canonical isomorphism: $$T^{k}V\otimes T^{\ell }V\to T^{k+\ell }V.$$
  • Note that $\otimes$ of $T(V)$ is not a tensor product.

• For $\alpha ,\beta \in \bigwedge (V)$, $$\alpha \wedge \beta =[\alpha \otimes \beta ]$$
• where $[\cdot ]$ denotes the equivalence classes with respect to $I$.

• Exterior product

• Intuitively: $A$ is a subobject of $B$ and $C$ is the correspoding factor object $B/A$
  • $$C\cong B/\mathrm{im}(f)=B/\ker (g)$$

• A short exact sequence is called split if there exists a homomorphism $h:C\to B$ such that $g\circ h={\mathrm{id}}_C$.
• If the objects are abelian groups, $B\cong A\oplus C$.