Quotient of Transformation Group acts Effectively

Let $G$ be a transformation group (which may or may not be effective) acting on $X$. Then the quotient group $G / G_0$, where $G_0$ is the kernel, ''does'' act effectively on $X$.


Condition for Group to Act Effectively on Left Coset Space

Let $G$ be a group whose identity is $e$. Let $H$ be a subgroup of $G$. Then $G$ acts effectively on the left coset space $G / H$ {{iff}}: :$\ds \bigcap_{a \mathop \in G} H^a = \set e$ where $H^a$ denotes the conjugate of $H$ by $a$.


Set of Permutations is Largest Effective Transformation Group

The set of permutations of a set $X$ forms the largest effective transformation group of $X$.


Conjugacy Action is Group Action

Let $\left({G, \circ}\right)$ be a group whose identity is $e$.


Conjugacy Action on Identity

Let $G$ be a group whose identity is $e$. For the conjugacy action: :$\order {\Orb e} = 1$ and thus: :$\Stab e = G$


Cauchy's Lemma (Group Theory)

Let $\struct {G, \circ}$ be a group of finite order whose identity is $e$. Let $p$ be a prime number which divides the order of $G$. Then $\struct {G, \circ}$ has an element of order $p$.


Transposition is Self-Inverse

All transpositions are self-inverse.


Conjugates of Transpositions

Let $k_1, k_2, k_3 \in \left\{{1, 2, \ldots, n}\right\}$. Then: : $(1): \quad \begin{bmatrix} k_1 & k_2 \end{bmatrix} = \begin{bmatrix} k_3 & k_2 \end{bmatrix} \begin{bmatrix} k_1 & k_3 \end{bmatrix} \begin{bmatrix} k_3 & k_2 \end{bmatrix}$ : $(2): \quad \begin{bmatrix} k_1 & k_2 \end{bmatrix} = \begin{bmatrix} k_3 & k_1 \end{bmatrix} \begin{bmatrix} k_3 & k_2 \end{bmatrix} \begin{bmatrix} k_3 & k_1 \end{bmatrix}$


K-Cycle can be Factored into Transpositions

Every $k$-cycle can be factorised into the product of $k - 1$ transpositions. This factorisation is not unique.


Sign of Permutation is Plus or Minus Unity

Let $n \in \N$ be a natural number. Let $\N_n$ denote the set of natural numbers $\set {1, 2, \ldots, n}$. Let $S_n$ denote the symmetric group on $n$ letters. Let $\sequence {x_k}_{k \mathop \in \N_n}$ be a finite sequence in $\R$. Let $\pi \in S_n$. Let $\map {\Delta_n} {x_1, x_2, \ldots, x_n}$ be the product of differences of $\tuple {x_1, x_2, \ldots, x_n}$. Let $\map \sgn \pi$ be the sign of $\pi$. Let $\pi \cdot \map {\Delta_n} {x_1, x_2, \ldots, x_n}$ be defined as: :$\pi \cdot \map {\Delta_n} {x_1, x_2, \ldots, x_n} := \map {\Delta_n} {x_{\map \pi 1}, x_{\map \pi 2}, \ldots, x_{\map \pi n} }$ Then either: :$\pi \cdot \Delta_n = \Delta_n$ or: :$\pi \cdot \Delta_n = -\Delta_n$ That is: :$\map \sgn \pi = \begin{cases} 1 & :\pi \cdot \Delta_n = \Delta_n \\ -1 & : \pi \cdot \Delta_n = -\Delta_n \end{cases}$ Thus: :$\pi \cdot \Delta_n = \map \sgn \pi \Delta_n$


Parity Function is Homomorphism

Let $S_n$ denote the symmetric group on $n$ letters. Let $\pi \in S_n$. Let $\map \sgn \pi$ be the sign of $\pi$. Let the parity function of $\pi$ be defined as: :Parity of $\pi = \begin{cases} \mathrm {Even} & : \map \sgn \pi = 1 \\ \mathrm {Odd} & : \map \sgn \pi = -1 \end{cases}$ The mapping $\sgn: S_n \to C_2$, where $C_2$ is the cyclic group of order 2, is a homomorphism.


Permutation on Polynomial is Group Action

Let $n \in \Z: n > 0$. Let $F_n$ be the set of all polynomials in $n$ variables $x_1, x_2, \ldots, x_n$: :$F = \set {\map f {x_1, x_2, \ldots, x_n}: f \text{ is a polynomial in $n$ variables} }$ Let $S_n$ denote the symmetric group on $n$ letters. Let $*: S_n \times F \to F$ be the mapping defined as: :$\forall \pi \in S_n, f \in F: \pi * \map f {x_1, x_2, \ldots, x_n} = \map f {x_{\map \pi 1}, x_{\map \pi 2}, \ldots, x_{\map \pi n} }$ Then $*$ is a group action.


Parity of Inverse of Permutation

Let $S_n$ denote the symmetric group on $n$ letters. Then: :$\forall \pi \in S_n: \map \sgn \pi = \map \sgn {\pi^{-1} }$


Parity of Conjugate of Permutation

Let $S_n$ denote the symmetric group on $n$ letters. Then: :$\forall \pi, \rho \in S_n: \map \sgn {\pi \rho \pi^{-1} } = \map \sgn \rho$ where $\map \sgn \pi$ is the sign of $\pi$.


Parity of K-Cycle

Let $\pi$ be a $k$-cycle. Then: :$\map \sgn \pi = \begin{cases} 1 & : k \ \text {odd} \\ -1 & : k \ \text {even} \end{cases}$ Thus: :$\map \sgn \pi = \paren {-1}^{k - 1}$ or equivalently: :$\map \sgn \pi = \paren {-1}^{k + 1}$


Alternating Group is Normal Subgroup of Symmetric Group

Let $n \ge 2$ be a natural number. Let $S_n$ denote the symmetric group on $n$ letters. Let $A_n$ be the alternating group on $n$ letters. Then $A_n$ is a normal subgroup of $S_n$ whose index is $2$.


Group of Permutations either All or Half Even

Let $G$ be a group of permutations. Then either ''exactly half'' of the permutations in $G$ are even, or they are ''all'' even.


Cycle Decomposition of Conjugate

Let $S_n$ denote the symmetric group on $n$ letters. Let $\pi, \rho \in S_n$. The cycle decomposition of the permutation $\pi \rho \pi^{-1}$ can be obtained from that of $\rho$ by replacing each $i$ in the cycle decomposition of $\rho$ with $\map \pi i$.


Conjugate Permutations have Same Cycle Type

Let $n \ge 1$ be a natural number. Let $G$ be a subgroup of the symmetric group on $n$ letters $S_n$. Let $\sigma, \rho \in G$. Then $\sigma$ and $\rho$ are conjugate {{iff}} they have the same cycle type.


Transpositions of Adjacent Elements generate Symmetric Group

Let $n \in \Z: n > 1$. Let $S_n$ denote the symmetric group on $n$ letters. Then the transpositions $a_k = \begin{pmatrix} k & k + 1 \end{pmatrix}$ for $1 \le k < n$ are a set of generators for $S_n$. They satisfy the relations: {{begin-eqn}} {{eqn | l = a_k^2       | r = e       | c = (for $1 \le k < n$) }} {{eqn | l = \paren {a_k a_{k + 1} }^3       | r = e       | c = (for $1 \le k < n - 1$) }} {{eqn | l = \paren {a_i a_j}^2       | r = e       | c = (for $1 \le i, j < n, \size {i - j} > 1$) }} {{end-eqn}}


Permutation of Cosets

Let $G$ be a group and let $H \le G$. Let $\mathbb S$ be the set of all distinct left cosets of $H$ in $G$. Then: :$(1): \quad$ For any $g \in G$, the mapping $\theta_g: \mathbb S \to \mathbb S$ defined by $\map {\theta_g} {x H} = g x H$ is a permutation of $\mathbb S$. :$(2): \quad$ The mapping $\theta$ defined by $\map \theta g = \theta_g$ is a homomorphism from $G$ into the symmetric group on $\mathbb S$. :$(3): \quad$ The kernel of $\theta$ is the subgroup $\displaystyle \bigcap_{x \mathop \in G} x H x^{-1}$.


Basic Results about Modules

Let $\struct {G, +_G}$ be an abelian group whose identity is $e$. Let $\struct {R, +_R, \times_R}$ be a ring whose zero is $0_R$. Let $\struct {G, +_G, \circ}_R$ be an $R$-module. Let $x \in G, \lambda \in R, n \in \Z$. Let $\sequence {x_m}$ be a sequence of elements of $G$. Let $\sequence {\lambda_m}$ be a sequence of elements of $R$ that is, scalars. Then:


Basic Results about Unitary Modules

Let $\struct {G, +_G}$ be an abelian group whose identity is $e$. Let $\struct {R, +_R, \times_R}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\struct {G, +_G, \circ}_R$ be an unitary $R$-module. Let $x \in G, n \in \Z$. Then:


Epimorphism preserves Modules

Let $\left({G, +_G, \circ}\right)_R$ be an $R$-module. Let $\left({H, +_H, \circ}\right)_R$ be an $R$-algebraic structure. Let $\phi: G \to H$ be an epimorphism. Then $H$ is an $R$-module.


Condition for Linear Transformation

Let $G$ be a unitary $R$-module, and let $H$ be an $R$-module. Let $\phi: G \to H$ be a mapping. Then $\phi$ is a linear transformation {{iff}}: :$\forall x, y \in G: \forall \lambda, \mu \in R: \map \phi {\lambda x + \mu y} = \lambda \map \phi x + \mu \map \phi y$


Module of All Mappings is Module

Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {G, +_G, \circ}_R$ be an $R$-module. Let $S$ be a set. Let $\struct {G^S, +_G', \circ}_R$ be the module of all mappings from $S$ to $G$. Then $\struct {G^S, +_G', \circ}_R$ is an $R$-module.


Kuratowski's Theorem

The following conditions on a graph $\Gamma$ are equivalent: :$(1): \quad \Gamma$ is planar :$(2): \quad \Gamma$ contains no subdivision of either the complete graph $K_5$ or the complete bipartite graph $K_{3, 3}$.


Fundamental Theorem of Finite Abelian Groups

Every finite abelian group is an internal group direct product of cyclic groups whose orders are prime powers. The number of terms in the product and the orders of the cyclic groups are uniquely determined by the group.


Finite Direct Product of Modules is Module

Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {G_1, +_1, \circ_1}_R, \struct {G_2, +_2, \circ_2}_R, \ldots, \struct {G_n, +_n, \circ_n}_R$ be $R$-modules. Let: :$\ds G = \prod_{k \mathop = 1}^n G_k$ be their direct product. Then $G$ is a module.


Subring Module

Let $\struct {R, +, \times}$ be a ring. Let $\struct {S, +_S, \times_S}$ be a subring of $R$. Let $\struct {G, +_G, \circ}_R$ be an $R$-module. Let $\circ_S$ be the restriction of $\circ$ to $S \times G$. Then $\struct {G, +_G, \circ_S}_S$ is an $S$-module. The module $\struct {G, +_G, \circ_S}_S$ is called the '''$S$-module obtained from $\struct {G, +_G, \circ}_R$ by restricting scalar multiplication'''. {{refactor|Extract the below into its own page}} If $\struct {G, +_G, \circ}_R$ is a unitary $R$-module and $1_R \in S$, then $\struct{G, +_G, \circ_S}_S$ is also unitary.


Projection on Cartesian Product of Modules

Let $G$ be the cartesian product of a sequence $\sequence {G_n}$ of $R$-modules. Then for each $j \in \closedint 1 n$, the projection $\pr_j$ on the $j$th co-ordinate is an epimorphism from $G$ onto $G_j$.


Submodule Test

Let $\left({G, +, \circ}\right)_R$ be a unitary $R$-module. Let $H$ be a non-empty subset of $G$. Then $\left({H, +, \circ}\right)_R$ is a submodule of $G$ iff: :$\forall x, y \in H: \forall \lambda \in R: x + y \in H, \lambda \circ x \in H$


Module is Submodule of Itself

Let $\left({G, +_G, \circ}\right)_R$ be an $R$-module. Then $\left({G, +_G, \circ}\right)_R$ is a submodule of itself.


Null Module Submodule of All

Let $\left({G, +_G, \circ}\right)_R$ be an $R$-module. Then the null module: :$\left({\left\{{e_G}\right\}, +_G, \circ}\right)_R$ is a submodule of $\left({G, +_G, \circ}\right)_R$.


First Sylow Theorem

Let $p$ be a prime number. Let $G$ be a group such that: :$\order G = k p^n$ where: :$\order G$ denotes the order of $G$ :$p$ is not a divisor of $k$. Then $G$ has at least one Sylow $p$-subgroup.


H-Cobordism Theorem

Let $X^n, Y^n$ be two simply connected manifolds. Let $n \in \N: n \ge 5$ and $\exists W$ such that $W$ is an h-cobordism between $X$ and $Y$. Then $\exists \psi: W \to X \times \closedint 0 1$ such that $\psi$ is a diffeomorphism. In particular, $X$ and $Y$ are diffeomorphic.


Group has Subgroups of All Prime Power Factors

Let $p$ be a prime. Let $G$ be a finite group of order $n$. If $p^k \divides n$ then $G$ has at least one subgroup of order $p^k$.


Normalizer of Sylow p-Subgroup

Let $P$ be a Sylow $p$-subgroup of a finite group $G$. Let $\map {N_G} P$ be the normalizer of $P$. Then any $p$-subgroup of $\map {N_G} P$ is contained in $P$. In particular, $P$ is the unique Sylow $p$-subgroup of $\map {N_G} P$.


Second Sylow Theorem

Let $P$ be a Sylow $p$-subgroup of the finite group $G$. Let $Q$ be any $p$-subgroup of $G$. Then $Q$ is a subset of a conjugate of $P$.


Third Sylow Theorem

All the Sylow $p$-subgroups of a finite group are conjugate.


Fourth Sylow Theorem

The number of Sylow $p$-subgroups of a finite group is congruent to $1 \pmod p$.


Sylow p-Subgroup is Unique iff Normal

A group $G$ has exactly one Sylow $p$-subgroup $P$ {{iff}} $P$ is normal.


Intersection of Normal Subgroup with Sylow P-Subgroup

Let $P$ be a Sylow $p$-subgroup of a finite group $G$. Let $N$ be a normal subgroup of $G$. Then $P \cap N$ is a Sylow $p$-subgroup of $N$.


Quotient of Sylow P-Subgroup

Let $P$ be a Sylow $p$-subgroup of a finite group $G$. Let $N$ be a normal subgroup of $G$. Then $P N / N$ is a Sylow $p$-subgroup of $G / N$.


Fifth Sylow Theorem

The number of Sylow $p$-subgroups of a finite group is a divisor of their common index.


Finite Submodule of Function Space

Let $\left({G, +}\right)$ be a group whose identity is $e$. Let $R$ be a ring. Let $\left({G, +, \circ}\right)_R$ be an $R$-module. Let $S$ be a set. Let $G^S$ the set of all mappings $f: S \to G$. Let $G^{\left({S}\right)}$ be the set of all mappings $f: S \to G$ such that $f \left({x}\right) = e$ for all but finitely many elements $x$ of $S$. Then: : $\left({G^{\left({S}\right)}, +', \circ}\right)_R$ is a submodule of $\left({G^S, +, \circ}\right)_R$ where $+'$ is the operation induced on $G^{\left({S}\right)}$ by $+$.


Polynomial Functions form Submodule of All Functions

Let $K$ be a commutative ring with unity. Let $K^K$ be the $K$-module mappings $f: K \to K$. Let $P \left({K}\right) \subseteq K^K$ be the set of all polynomial functions on $K$. Then $P \left({K}\right)$ is a $K$-submodule of $K^K$.


Intersection and Sum of Submodules

Let $\left({G, +, \circ}\right)_R$ be an $R$-module. Let $H$ and $K$ be submodules of $G$. Then $H + K$ and $H \cap K$ are also submodules of $G$. The intersection of any set of submodules of $G$ is a submodule. Thus if $S \subseteq G$, the intersection of all submodules of $G$ containing $S$ is the smallest submodule of $G$ containing $S$.


Linear Transformation of Submodule

Let $G$ and $H$ be $R$-modules. Let $\phi: G \to H$ be a linear transformation. Then: :$(1): \quad$ If $M$ is a submodule of $G$, $\phi \sqbrk M$ is a submodule of $H$ :$(2): \quad$ If $N$ is a submodule of $H$, $\phi^{-1} \sqbrk N$ is a submodule of $G$ :$(3): \quad$ The codomain of $\phi$ is a submodule of $H$ :$(4): \quad$ The kernel of $\phi$ is a submodule of $G$.


Poincaré Conjecture

Let $\Sigma^m$ be a smooth $m$-manifold. Let $\Sigma^m$ satisfy: :$H_0 \struct {\Sigma; \Z} = 0$ and: :$H_m \struct {\Sigma; \Z} = \Z$ {{explain|Definition of the notation $H_0 \struct {\Sigma; \Z}$, nature of $H_0$ and $H_m$}} Then $\Sigma^m$ is homeomorphic to the $m$-sphere $\Bbb S^m$.


Linear Transformation of Generated Module

Let $G$ and $H$ be $R$-modules. Let $\phi$ and $\psi$ be linear transformations $G$ into $H$.  Let $S$ be a generator for $G$. Suppose that $\forall x \in S: \map \phi x = \map \psi x$. Then $\phi = \psi$.


Generated Submodule is Linear Combinations

Let $G$ be a unitary $R$-module. Let $S \subseteq G$. Then the submodule $H$ generated by $S$ is the set of all linear combinations of $S$.


Empty Set is Linearly Independent

The empty set is a linearly independent set.


Subset of Module Containing Identity is Linearly Dependent

Let $G$ be a group whose identity is $e$. Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. Let $\struct {G, +_G, \circ}_R$ be an $R$-module. Let $H \subseteq G$ such that $e \in H$. Then $H$ is a linearly dependent set.


Subset of Linearly Independent Set is Linearly Independent

A subset of a linearly independent set is also linearly independent.


Superset of Linearly Dependent Set

Any set containing a linearly dependent set is also linearly dependent.


Number of Ordered Bases from a Basis

Each basis of $n$ elements determines $n!$ ordered bases.


Classification of Compact Two-Manifolds

Any smooth, compact, path-connected manifold of dimension $2$ is diffeomorphic to the sphere $\mathbb S^2$, a connected sum of tori $\mathbb T^2$, or a connected sum of projective spaces $\mathbb{RP}^2$. Any such $2$-manifold with boundary is diffeomorphic to the sphere $\mathbb S^2$, a connected sum of tori $\mathbb T^2$, or a connected sum of projective spaces $\mathbb{RP}^2$, with a number of open disks removed. The Euler characteristic, orientability, and number of boundary curves suffice to describe a surface.


Standard Ordered Basis is Basis

Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $n$ be a positive integer. For each $j \in \closedint 1 n$, let $e_j$ be the ordered $n$-tuple of elements of $R$ whose $j$th entry is $1_R$ and all of whose other entries is $0_R$. Then $\sequence {e_n}$ is an ordered basis of the $R$-module $R^n$. This ordered basis is called the '''standard ordered basis of $R^n$'''. The corresponding set $\set {e_1, e_2, \ldots, e_n}$ is called the '''standard basis of $R^n$'''.


Unique Representation by Ordered Basis

Let $G$ be a unitary $R$-module. Then $\sequence {a_n}$ is an ordered basis of $G$ {{Iff}}: :For every $x \in G$ there exists one and only one sequence $\sequence {\lambda_n}$ of scalars such that $\displaystyle x = \sum_{k \mathop = 1}^n \lambda_k a_k$.


Isomorphism from R^n via n-Term Sequence

Let $G$ be a unitary $R$-module. Let $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ be an ordered basis of $G$. Let $R^n$ be the $R$-module $R^n$. Let $\psi: R^n \to G$ be defined as: :$\displaystyle \map \psi {\sequence {\lambda_k}_{1 \mathop \le k \mathop \le n} } = \sum_{k \mathop = 1}^n \lambda_k a_k$ Then $\psi$ is an isomorphism.


Unitary R-Modules with n-Element Bases Isomorphic

Any two unitary $R$-modules having bases of $n$ elements are isomorphic.


R-Module R^n is n-Dimensional

The $R$-module $R^n$ is $n$-dimensional.


Unique Linear Transformation Between Modules

Let $G$ and $H$ be unitary $R$-modules. Let $\left \langle {a_n} \right \rangle$ be an ordered basis of $G$. Let $\left \langle {b_n} \right \rangle$ be a sequence of elements of $H$. Then there is a unique linear transformation $\phi: G \to H$ satisfying $\forall k \in \left[{1 \,.\,.\, n}\right]: \phi \left({a_k}\right) = b_k$


Linear Transformation from Ordered Basis less Kernel

Let $G$ and $H$ be unitary $R$-modules. Let $\phi: G \to H$ be a non-zero linear transformation. Let $G$ be $n$-dimensional. Let $\left \langle {a_n} \right \rangle$ be any ordered basis of $G$ such that $\left\{{a_k: r + 1 \le k \le n}\right\}$ is the basis of the kernel of $\phi$. Then $\left \langle {\phi \left({a_r}\right)} \right \rangle$ is an ordered basis of the image of $\phi$.


Addition of Linear Transformations

Let $\left({G, +_G, \circ}\right)_R$ and $\left({H, +_H, \circ}\right)_R$ be $R$-modules. Let $\phi: G \to H$ and $\psi: G \to H$ be linear transformations. Let $\phi +_H \psi$ be the operation on $H^G$ induced by $+_H$ as defined in Induced Structure. Then $\phi +_H \psi: G \to H$ is a linear transformation.


Negative Linear Transformation

Let $\left({G, +_G, \circ}\right)_R$ and $\left({H, +_H, \circ}\right)_R$ be $R$-modules. Let $\phi: G \to H$ be a linear transformation. Let $- \phi$ be the negative of $\phi$ as defined in Induced Structure Inverse. Then $- \phi: G \to H$ is also a linear transformation.


Morse-Sard Theorem

Let $f: X \to Y$ be any smooth map of manifolds. Then almost every point in $Y$ is a regular value of $f$.


Linear Transformation from Center of Scalar Ring

Let $\struct {G, +_G, \circ}_R$ and $\struct {H, +_H, \circ}_R$ be $R$-modules. Let $\phi: G \to H$ be a linear transformation. Let $\map Z R$ be the center of the scalar ring $R$. Let $\lambda \in \map Z R$. Then $\lambda \circ \phi$ is a linear transformation.


Product of Linear Transformations

Let $R$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\struct {G, +_G, \circ}_R$ be a unitary $R$-module such that $\map \dim G = n$. Let $\struct {H, +_H, \circ}_R$ be a unitary $R$-module such that $\map \dim H = m$. Let $\map {\LL_R} {G, H}$ be the set of all linear transformations from $G$ to $H$. Then: :$\map \dim {\map {\LL_R} {G, H} } = n m$ Let $\sequence {a_n}$ be an ordered basis for $G$. Let $\sequence {b_m}$ be an ordered basis for $H$. Let $\phi_{i j}: G \to H$ be the unique linear transformation defined for each $i \in \closedint 1 n, j \in \closedint 1 m$ which satisfies: :$\forall k \in \closedint 1 n: \map {\phi_{i j} } {a_k} = \delta_{i k} b_j$ where $\delta$ is the Kronecker delta. Then: :$\set {\phi_{i j}: i \in \closedint 1 n, j \in \closedint 1 m}$ is a basis for $\map \dim {\map {\LL_R} {G, H} }$.


Dimension of Algebraic Dual

Let $G$ be an $n$-dimensional $R$-module. Let $G^*$ be the algebraic dual of $G$. Let $G^{**}$ be the algebraic dual of $G^*$. Then $G^*$ and $G^{**}$ are also $n$-dimensional.


Annihilator is Submodule of Algebraic Dual

Let $R$ be a commutative ring. Let $G$ be a module over $R$. Let $M$ be a submodule of $G$. Let $G^*$ be the algebraic dual of $G$. Then the annihilator $M^\circ$ of $M$ is a submodule of $G^*$. Similarly, let $N$ be a submodule of $G^*$. Let $G^{**}$ be the algebraic dual of $G^*$. Then the annihilator $N^\circ$ of $N$ is a submodule of $G^{**}$.


Properties of Evaluation Linear Transformation

Let $R$ be a commutative ring. Let $G$ be an $R$-module. Let $G^*$ be the algebraic dual of $G$. Let $\left \langle {x, t'} \right \rangle$ be the evaluation linear transformation from $G$ to $G^{**}$. Then the mapping $\phi: G \times G^* \to R$ defined as $\forall \left({x, t'}\right) \in G \times G^*: \phi \left({x, t'}\right) = \left \langle {x, t'} \right \rangle$ satisfies the following properties: : $(1): \quad \forall x, y \in G: \forall t' \in G^*: \left \langle {x + y, t'} \right \rangle = \left \langle {x, t'} \right \rangle + \left \langle {y, t'} \right \rangle$ : $(2): \quad \forall x \in G: \forall s', t' \in G^*: \left \langle {x, s' + t'} \right \rangle = \left \langle {x, s'} \right \rangle + \left \langle {x, t'} \right \rangle$ : $(3): \quad \forall x \in G: \forall s', t' \in G^*: \forall \lambda \in R: \left \langle {\lambda x, t'} \right \rangle = \lambda \left \langle {x, t'} \right \rangle = \left \langle {x, \lambda t'} \right \rangle$


Zero Vector Space Product iff Factor is Zero

Let $F$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $\struct {\mathbf V, +, \circ}_F$ be a vector space over $F$, as defined by the vector space axioms. Let $\mathbf v \in \mathbf V, \lambda \in F$. Then: :$\lambda \circ \mathbf v = \bszero \iff \paren {\lambda = 0_F \lor x = \bszero}$


Homomorphic Image of Vector Space

Let $\struct {K, +_K, \times_K}$ be a division ring. Let $\struct {V, +_V, \circ_V}_K$ be a $K$-vector space. Let $\struct {W, +_W, \circ_W}_K$ be a $K$-algebraic structure. Let $\phi: V \to W$ be a homomorphism, i.e. a linear transformation. Then the homomorphic image of $\phi$ is a $K$-vector space.


Direct Product of Vector Spaces is Vector Space

Let $K$ be a field. Let $V_1, V_2, \ldots, V_n$ be $K$-vector spaces. Let $\struct {V, + , \circ}_K$ be their direct product. Then $\struct {V, + , \circ}_K$ is a $K$-vector space.


Definition:Vector Space on Cartesian Product

Let $\struct {K, +, \circ}$ be a division ring. Let $n \in \N_{>0}$. Let $+: K^n \times K^n \to K^n$ be defined as: :$\tuple {\alpha_1, \ldots, \alpha_n} + \tuple {\beta_1, \ldots, \beta_n} = \tuple {\alpha_1 +_R \beta_1, \ldots, \alpha_n +_R \beta_n}$ Let $\times: K \times K^n \to K^n$ be defined as: :$\lambda \times \tuple {\alpha_1, \ldots, \alpha_n} = \tuple {\lambda \times_R \alpha_1, \ldots, \lambda \times_R \alpha_n}$ Then $\struct {K^n, +, \times}_K$ is '''the $K$-vector space $K^n$'''.


Definition:Vector Space over Division Subring

Let $\struct {R, +, \circ}$ be a ring with unity whose unity is $1_R$. Let $S$ be a division subring of $R$, such that $1_R \in S$. Then $\struct {R, +, \circ_S}_S$, where $\circ_S$ is the restriction of $\circ$ to $S \times R$, is the '''vector space on $R$ over the division subring $S$'''.


Division Ring is Vector Space over Prime Subfield

Let $\struct {K, +, \times}$ be a division ring. Let $\struct {S, +, \times}$ be the prime subfield of $K$ Then $\struct {K, +, \times_S}_S$ is an $S$-vector space, where $\times_S$ is the restriction of $\times$ to $S \times K$.


Vector Space over Subring

Let $K$ be a division subring of the division ring $\struct {L, +_L, \times_L}$. Let $\struct {G, +_G, \circ}_L$ be a $L$-vector space. Then $\struct {G, +_G, \circ_K}_K$ is a $K$-vector space, where $\circ_K$ is the restriction of $\circ$ to $K \times G$. The $K$-vector space $\struct {G, +_G, \circ_K}_K$ is called the '''$K$-vector space obtained from $\struct {L, +_L, \times_L}$ by restricting scalar multiplication'''.


Subspaces of Dimension 2 Real Vector Space

Take the $\R$-vector space $\left({\R^2, +, \times}\right)_\R$. Let $S$ be a subspace of $\left({\R^2, +, \times}\right)_\R$. Then $S$ is one of: : $(1): \quad \left({\R^2, +, \times}\right)_\R$ : $(2): \quad \left\{{0}\right\}$ : $(3): \quad$ A line through the origin.


Subspace of Real Continuous Functions

Let $\mathbb J = \set {x \in \R: a \le x \le b}$ be a closed interval of the real number line $\R$. Let $\map \CC {\mathbb J}$ be the set of all continuous real functions on $\mathbb J$. Then $\struct {\map \CC {\mathbb J}, +, \times}_\R$ is a subspace of the $\R$-vector space $\struct {\R^{\mathbb J}, +, \times}_\R$.


Singleton is Linearly Independent

Let $K$ be a division ring. Let $\struct {G, +_G}$ be a group whose identity is $e$. Let $\struct {G, +_G, \circ}_K$ be a $K$-vector space whose zero is $0_K$. Let $x \in G: x \ne e$. Then $\set x$ is a linearly independent subset of $G$.


Linearly Dependent Sequence of Vector Space

Let $\struct {G, +}$ be a group whose identity is $\mathbf 0$. Let $\struct {G, +, \circ}_K$ be a $K$-vector space. Let $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ be a sequence of distinct non-zero vectors of $G$. Then $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ is linearly dependent {{iff}}: :$\exists p \in \closedint 2 n: a_p$ is a linear combination of $\sequence {a_k}_{1 \mathop \le k \mathop \le p - 1}$


Vector Space has Basis Between Linearly Independent Set and Finite Spanning Set

Let $K$ be a division ring. Let $G$ be a finitely generated $K$-vector space. Let $H$ be a linearly independent subset of $G$. Let $F$ be a finite generator for $G$ such that $H \subseteq F$. Then there is a basis $B$ for $G$ such that $H \subseteq B \subseteq F$.


Bases of Finitely Generated Vector Space have Equal Cardinality

Let $K$ be a division ring. Let $G$ be a finitely generated $K$-vector space. Then any two bases of $G$ are finite and equivalent. {{improve|I still think there's a simpler way of saying "they have the same number of elements" than bringing in all that top-heavy technical set theoretic language of "set equivalence" and "cardinality" and so on. Since we are talking about a finite set, the complexities which arise with regard to transfinites do not arise, so there is no direct need to go into such depth.}}


Linearly Independent Subset also Independent in Generated Subspace

Let $G$ be a finitely generated $K$-vector space. Let $S$ be a linearly independent subset of $G$. Let $M$ be the subspace of $G$ generated by $S$. If $M \ne G$, then $\forall b \in G: b \notin M$, the set $S \cup \set b$ is linearly independent.


Sufficient Conditions for Basis of Finite Dimensional Vector Space

Let $K$ be a division ring. Let $n \ge 0$ be a natural number. Let $E$ be an $n$-dimensional vector space over $K$. Let $B \subseteq E$ be a subset such that $\card B = n$. {{TFAE}} : $(1): \quad$ $B$ is a basis of $E$. : $(2): \quad$ $B$ is linearly independent. : $(3): \quad$ $B$ is a generator for $E$.


Dimension of Proper Subspace is Less Than its Superspace

Let $G$ be a vector space whose dimension is $n$. Let $H$ be a subspace of $G$. Then $H$ is finite dimensional, and $\map \dim H \le \map \dim G$. If $H$ is a proper subspace of $G$, then $\map \dim H < \map \dim G$.


Results concerning Generators and Bases of Vector Spaces

Let $E$ be a vector space of $n$ dimensions. Let $G$ be a generator for $E$. Then $G$ has the following properties:


Grassmann's Identity

Let $K$ be a division ring. Let $\struct {G, +_G, \circ}_K$ be a $K$-vector space. Let $M$ and $N$ be finite-dimensional subspaces of $G$. Then the sum $M + N$ and intersection $M \cap N$ are finite-dimensional, and: :$\map \dim {M + N} + \map \dim {M \cap N} = \map \dim M + \map \dim N$


Rank Plus Nullity Theorem

Let $G$ be an $n$-dimensional vector space. Let $H$ be a vector space. Let $\phi: G \to H$ be a linear transformation. Let $\map \rho \phi$ and $\map \nu \phi$ be the rank and nullity respectively of $\phi$. Then the image of $\phi$ is finite-dimensional, and: :$\map \rho \phi + \map \nu \phi = n$ By definition of rank and nullity, it can be seen that this is equivalent to the alternative way of stating this result: :$\map \dim {\Img \phi} + \map \dim {\map \ker \phi} = \map \dim G$ {{wtd|and the theorem is applicable to matrices}}


Linear Transformation of Vector Space Monomorphism

Let $G$ and $H$ be a $K$-vector space. Let $\phi: G \to H$ be a linear transformation. Then $\phi$ is a monomorphism {{iff}} for every linearly independent sequence $\sequence {a_n}$ of vectors of $G$, $\sequence {\map \phi {a_n} }$ is a linearly independent sequence of vectors of $H$.


Linear Transformation of Vector Space Equivalent Statements

Let $G$ and $H$ be $n$-dimensional vector spaces. Let $\phi: G \to H$ be a linear transformation. Then these statements are equivalent: : $(1): \quad \phi$ is an isomorphism. : $(2): \quad \phi$ is a monomorphism. : $(3): \quad \phi$ is an epimorphism. : $(4): \quad \phi \left({B}\right)$ is a basis of $H$ for every basis $B$ of $G$. : $(5): \quad \phi \left({B}\right)$ is a basis of $H$ for some basis $B$ of $G$.


Results Concerning Annihilator of Vector Subspace

Let $G$ be an $n$-dimensional vector space over a field. Let $J: G \to G^{**}$ be the evaluation isomorphism. Let $G^*$ be the algebraic dual of $G$. Let $G^{**}$ be the algebraic dual of $G^*$. Let $M$ be an $m$-dimensional subspace of $G$. Let $N$ be a $p$-dimensional subspace of $G^*$. Let $M^\circ$ be the annihilator of $M$. Then: :$(1): \quad M^\circ$ is an $\paren {n - m}$-dimensional subspace of $G^*$, and $M^{\circ \circ} = \map J M$ :$(2): \map {\quad J^{-1} } {N^\circ}$ is an $\paren {n - p}$-dimensional subspace of $G$ :$(3): \quad$ The mapping $M \to M^\circ$ is a bijection from the set of all $m$-dimensional subspaces of $G$ onto the set of all $\paren {n - m}$-dimensional subspaces of $G^*$ :$(4): \quad$ Its inverse is the bijection $N \to \map {J^{-1} } {N^\circ}$.


Vector Subspace Dimension One Less

Let $K$ be a field. Let $M$ be a subspace of the $n$-dimensional vector space $K^n$. The following statements are equivalent: :$(1): \quad \map \dim M = n - 1$ :$(2): \quad M$ is the kernel of a nonzero linear form :$(3): \quad$ There exists a sequence $\sequence {\alpha_n} $ of scalars, not all of which are zero, such that: :::$M = \set {\tuple {\lambda_1, \ldots, \lambda_n} \in K^n: \alpha_1 \lambda_1 + \cdots + \alpha_n \lambda_n = 0}$ Also, suppose the above hold. Let $\sequence {\beta_n}$ be a sequence of scalars such that: :$M = \set {\tuple {\lambda_1, \ldots, \lambda_n} \in K^n: \beta_1 \lambda_1 + \cdots + \beta_n \lambda_n = 0}$ Then there is a non-zero scalar $\gamma$ such that: :$\forall k \in \closedint 1 n: \beta_k = \gamma \alpha_k$


Rank and Nullity of Transpose

Let $G$ and $H$ be $n$-dimensional vector spaces over a field. Let $\mathcal L \left({G, H}\right)$ be the set of all linear transformations from $G$ to $H$. Let $u \in \mathcal L \left({G, H}\right)$. Let $u^t$ be the transpose of $u$. Then: : $(1): \quad$ $u$ and $u^t$ have the same rank and nullity : $(2): \quad$ $\ker \left({u^t}\right)$ is the annihilator of the image of $u$ : $(3): \quad$ The image of $u^t$ is the annihilator of $\ker \left({u}\right)$.


Linear Operator on the Plane

Let $\phi$ be a linear operator on the real vector space of two dimensions $\R^2$. Then $\phi$ is completely determined by an ordered tuple of $4$ real numbers.


Similarity Mapping of Plane is Linear Operator

Let $G$ be a vector space over a field $K$. Let $\beta \in K$. Then the mapping: :$s_\beta: G \to G$ defined by $\map {s_\beta} {\mathbf x} = \beta \mathbf x$ is a linear operator on $G$. If $\beta \ne 0$ then $s_\beta$ is an automorphism of $G$, and $\paren {s_\beta}^{-1} = s_{\beta^{-1} }$ The linear operators $s_\beta$, where $\beta \ne 0$, are called '''similarities of $G$'''.


Cantor-Dedekind Hypothesis

The points on an infinite straight line are in one-to-one correspondence with the set $\R$ of real numbers. Hence the set of all points on an infinite straight line and $\R$ are equinumerous.


Full Angle measures 2 Pi Radians

One full angle is equal to $2 \pi$ radians. :$2 \pi \approx 6 \cdotp 28318 \, 53071 \, 79586 \, 4769 \ldots$ {{OEIS|A019692}}


Rotation of Plane about Origin is Linear Operator

Let $r_\alpha$ be the plane rotation of the plane about the origin through an angle of $\alpha$. That is, let $r_\alpha: \R^2 \to \R^2$ be the mapping defined as: :$\forall x \in \R^2: \map {r_\alpha} x = \text { the point into which a rotation of } \alpha \text{ carries } x$ Then $r_\alpha$ is a linear operator determined by the ordered sequence: : $\tuple {\cos \alpha  -\sin \alpha, \sin \alpha + \cos \alpha}$


Stretching and Contraction Mappings of Plane are Linear Operators

Let $s_\beta: \R^2 \to \R^2$ be a similarity of $\R^2$. Then $s_{-1}$ is the same as the rotation $r_{\pi}$ of the plane about the origin one half turn. If $\beta \ge 1$, then $s_\beta$ is called a '''stretching''', and if $0 < \beta \le 1$, $s_\beta$ is called a '''contraction'''. If $\beta < 0$, then $s_\beta$ is a stretching or contraction followed by a rotation one half turn. It is also the same as a rotation one half turn followed by a stretching or contraction.


Reflection of Plane in Line through Origin is Linear Operator

Let $M$ be a straight line in the plane passing through the origin. Then the '''reflection''' $s_M$ of $\R^2$ in $M$ is the rotation of the plane in space through one half turn about $M$ as an axis. :$s_M \circ s_M = I_{\R^2}$ and hence: :$s_M = s_M^{-1}$ If $M$ is the $x$-axis then $\map {s_M} {\lambda_1, \lambda_2} = \tuple {\lambda_1, -\lambda_2}$. If $M$ is the $y$-axis then $\map {s_M} {\lambda_1, \lambda_2} = \tuple {-\lambda_1, \lambda_2}$. In general, $s_M$ is a linear operator for every straight line $M$ through the origin.


Projection of Straight Line on Another in Plane

Let $M$ and $N$ be distinct straight lines through the plane through the origin. Let $\operatorname{pr}_{M, N}$ be the projection on $M$ along $N$. $M$ and $N$ are respectively the codomain and kernel of $\operatorname{pr}_{M, N}$. {{explain|As the kernel is a concept defined in relation to a homomorphism, it needs to be clarified what homomorphism is being considered.}} :$\operatorname{pr}_{M, N} \left({x}\right) = x \iff x \in M$ If $M$ is the $x$-axis and $N$ is the $y$-axis, then $\operatorname{pr}_{M, N} \left({\lambda_1, \lambda_2}\right) = \left({\lambda_1, 0}\right)$. If $M$ is the $y$-axis and $N$ is the $x$-axis, then $\operatorname{pr}_{M, N} \left({\lambda_1, \lambda_2}\right) = \left({0, \lambda_2}\right)$. Any such projection is a linear operator.


Condition for Straight Lines in Plane to be Parallel

Let $L: \alpha_1 x + \alpha_2 y = \beta$ be a straight line in $\R^2$. Then the straight line $L'$ is parallel to $L$ iff there is a $\beta' \in \R^2$ such that: :$L' = \set {\tuple {x, y} \in \R^2: \alpha_1 x + \alpha_2 y = \beta'}$


Equation of Plane

A plane $P$ is the set of all $\tuple {x, y, z} \in \R^3$, where: :$\alpha_1 x + \alpha_2 y + \alpha_3 z = \gamma$ where $\alpha_1, \alpha_2, \alpha_3, \gamma \in \R$ are given, and not all of $\alpha_1, \alpha_2, \alpha_3$ are zero.


Planes are Subspaces of Space

The two-dimensional subspaces of $\R^3$ are precisely the homogeneous planes of solid analytic geometry.


Matrix Space Semigroup under Hadamard Product

Let $\map {\MM_S} {m, n}$ be the matrix space over a semigroup $\struct {S, \cdot}$. Then the algebraic structure $\struct {\map {\MM_S} {m, n}, \circ}$, where $\circ$ is the Hadamard product, is also a semigroup. If $\struct {S, \cdot}$ is a commutative semigroup then so is $\struct {\map {\MM_S} {m, n}, \circ}$. If $\struct {S, \cdot}$ is a monoid then so is $\struct {\map {\MM_S} {m, n}, \circ}$.


Hadamard Product over Group forms Group

Let $\struct {G, \cdot}$ be a group whose identity is $e$. Let $\map {\MM_G} {m, n}$ be a $m \times n$ matrix space over $\struct {G, \cdot}$. Then $\struct {\map {\MM_G} {m, n}, \circ}$, where $\circ$ is Hadamard product, is also a group.


Matrix Multiplication is Associative

Let $R$ be a ring. Matrix multiplication (conventional) is associative.


Linear Transformations Isomorphic to Matrix Space

Let $R$ be a ring with unity. Let $F$, $G$ and $H$ be free $R$-modules of finite dimension $p,n,m>0$ respectively. Let $\left \langle {a_p} \right \rangle$, $\left \langle {b_n} \right \rangle$ and $\left \langle {c_m} \right \rangle$ be ordered bases  Let $\operatorname{Hom} \left({G, H}\right)$ be the set of all linear transformations from $G$ to $H$. Let $\mathcal M_R \left({m, n}\right)$ be the $m \times n$ matrix space over $R$. Let $\left[{u; \left \langle {c_m} \right \rangle, \left \langle {b_n} \right \rangle}\right]$ be the matrix of $u$ relative to $\left \langle {b_n} \right \rangle$ and $\left \langle {c_m} \right \rangle$. Let $M: \operatorname{Hom} \left({G, H}\right) \to \mathcal M_R \left({m, n}\right)$ be defined as: :$\forall u \in \operatorname{Hom} \left({G, H}\right): M \left({u}\right) = \left[{u; \left \langle {c_m} \right \rangle, \left \langle {b_n} \right \rangle}\right]$ Then $M$ is an isomorphism of modules, and: :$\forall u \in \operatorname{Hom} \left({F, G}\right), v \in \operatorname{Hom} \left({G, H}\right): \left[{v \circ u; \left \langle {c_m} \right \rangle, \left \langle {a_p} \right \rangle}\right] = \left[{v; \left \langle {c_m} \right \rangle, \left \langle {b_n} \right \rangle}\right] \left[{u; \left \langle {b_n} \right \rangle, \left \langle {a_p} \right \rangle}\right]$


Matrix Multiplication Distributes over Matrix Addition

Matrix multiplication (conventional) is distributive over matrix entrywise addition.


Unit Matrix is Unity of Ring of Square Matrices

Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\struct {\map {\MM_R} n, +, \times}$ denote the ring of square matrices of order $n$ over $R$. The unit matrix over $R$: :$\mathbf I_n = \begin {pmatrix} 1_R & 0_R & 0_R & \cdots & 0_R \\ 0_R & 1_R & 0_R & \cdots & 0_R \\ 0_R & 0_R & 1_R & \cdots & 0_R \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0_R & 0_R & 0_R & \cdots & 1_R \end {pmatrix}$ is the identity element of $\struct {\map {\MM_R} n, +, \times}$.


Matrix Multiplication is Closed

Let $\struct {R, +, \circ}$ be a ring. Let $\map {\MM_R} n$ be a $n \times n$ matrix space over $R$. Then matrix multiplication (conventional) over $\map {\MM_R} n$ is closed.


Square Matrices over Real Numbers under Multiplication form Monoid

Let $\map {\mathcal M_\R} n$ be a $n \times n$ matrix space over the set of real numbers $\R$. Then the set of all $n \times n$ real matrices $\map {\mathcal M_\R} n$ under matrix multiplication (conventional) forms a monoid.


Ring of Square Matrices over Commutative Ring with Unity

Let $R$ be a commutative ring with unity. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\struct {\map {\MM_R} n, +, \times}$ denote the ring of square matrices of order $n$ over $R$. Then $\struct {\map {\MM_R} n, +, \times}$ is a ring with unity. However, for $n \ge 2$, $\struct {\map {\MM_R} n, +, \times}$ is not a commutative ring.


Invertible Matrix corresponds to Automorphism

Let $R$ be a ring with unity. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $G$ be an $n$-dimensional $R$-module. Let $\map {\mathcal M_R} n$ be the $n \times n$ matrix space over $R$. Let $\map {\mathcal L_R} G$ be the set of all linear operators on $G$. Then the invertible elements of the ring of square matrices $\struct {\map {\mathcal M_R} n, +, \times}$ correspond directly to automorphisms of $\map {\mathcal L_R} G$.


Change of Basis is Invertible

Let $R$ be a ring with unity. Let $M$ be a free $R$-module of finite dimension $n>0$. Let $\mathcal A$ and $\mathcal B$ be ordered bases of $M$. Let $\mathbf P$ be the change of basis matrix from $\mathcal A$ to $\mathcal B$. Then $\mathbf P$ is invertible, and its inverse $\mathbf P^{-1}$ is the change of basis matrix from $\mathcal B$ to $\mathcal A$.


Matrix Corresponding to Change of Basis under Linear Transformation

Let $R$ be a ring with unity. Let $G$ and $H$ be free $R$-modules of finite dimensions $n,m>0$ respectively. Let $\left \langle {a_n} \right \rangle$ and $\left \langle {{a_n}'} \right \rangle$ be ordered bases of $G$. Let $\left \langle {b_m} \right \rangle$ and $\left \langle {{b_m}'} \right \rangle$ be ordered bases of $H$. Let $u: G \to H$ be a linear transformation, and let $\left[{u; \left \langle {b_m} \right \rangle, \left \langle {a_n} \right \rangle}\right]$ be the matrix of $u$ relative to $\left \langle {a_n} \right \rangle$ and $\left \langle {b_m} \right \rangle$. Let: : $\mathbf A = \left[{u; \left \langle {b_m} \right \rangle, \left \langle {a_n} \right \rangle}\right]$ : $\mathbf B = \left[{u; \left \langle {{b_m}'} \right \rangle, \left \langle {{a_n}'} \right \rangle}\right]$ Then: :$\mathbf B = \mathbf Q^{-1} \mathbf A \mathbf P$ where: : $\mathbf P$ is the matrix corresponding to the change of basis from $\left \langle {a_n} \right \rangle$ to $\left \langle {{a_n}'} \right \rangle$ : $\mathbf Q$ is the matrix corresponding to the change of basis from $\left \langle {b_m} \right \rangle$ to $\left \langle {{b_m}'} \right \rangle$.


Matrix Equivalence is Equivalence Relation

Matrix equivalence is an equivalence relation.


Equivalent Matrices have Equal Rank

Let $\mathbf A$ and $\mathbf B$ be $m \times n$ matrices over a field $K$. Let $\map \phi {\mathbf A}$ denote the rank of $\mathbf A$. Let $\mathbf A \equiv \mathbf B$ denote that $\mathbf A$ and $\mathbf B$ are matrix equivalent. Then: :$\mathbf A \equiv \mathbf B$ {{iff}}: :$\map \phi {\mathbf A} = \map \phi {\mathbf B}$


Transpose of Matrix Product

Let $\mathbf A$ and $\mathbf B$ be matrices over a commutative ring such that $\mathbf A \mathbf B$ is defined. Then $\mathbf B^\intercal \mathbf A^\intercal$ is defined, and: :$\paren {\mathbf A \mathbf B}^\intercal = \mathbf B^\intercal \mathbf A^\intercal$ where $\mathbf X^\intercal$ is the transpose of $\mathbf X$.


General Linear Group is Group

Let $K$ be a field. Let $\GL {n, K}$ be the general linear group of order $n$ over $K$. Then $\GL {n, K}$ is a group.


Transpose of Row Matrix is Column Matrix

Let $\mathbf x = \sqbrk x_{1 n} = \begin {bmatrix} x_1 & x_2 & \cdots & x_n \end {bmatrix}$ be a row matrix. Then $\mathbf x^\intercal$, the transpose of $\mathbf x$, is a column matrix: :$\begin {bmatrix} x_1 & x_2 & \cdots & x_n \end{bmatrix}^\intercal = \begin {bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end {bmatrix}$


Transpose of Transpose of Matrix

Let $\mathbf A$ be a matrix. Let $\mathbf A^\intercal$ be the transpose of $\mathbf A$. Then: :$\paren {\mathbf A^\intercal}^\intercal = \mathbf A$


Solution to Simultaneous Linear Equations

Let $\displaystyle \forall i \in \closedint 1 m: \sum _{j \mathop = 1}^n {\alpha_{i j} x_j} = \beta_i$ be a system of simultaneous linear equations. where all of $\alpha_1, \ldots, a_n, x_1, \ldots x_n, \beta_i, \ldots, \beta_m$ are elements of a field $K$. Then $x = \tuple {x_1, x_2, \ldots, x_n}$ is a solution of this system {{iff}}: :$\sqbrk \alpha_{m n} \sqbrk x_{n 1} = \sqbrk \beta_{m 1}$ where $\sqbrk a_{m n}$ is an $m \times n$ matrix.


Infinite Cyclic Group is Isomorphic to Integers

Let $G$ be an infinite cyclic group. Then $G$ is isomorphic to the additive group of integers: $G \cong \struct {\Z, +}$.


Subgroup of Infinite Cyclic Group is Infinite Cyclic Group

Let $G = \gen a$ be an infinite cyclic group generated by $a$, whose identity is $e$. Let $g \in G, g \ne e: \exists k \in \Z, k \ne 0: g = a^k$. Let $H = \gen g$. Then $H \le G$ and $H \cong G$. Thus, all non-trivial subgroups of an infinite cyclic group are themselves infinite cyclic groups. A subgroup of $G = \gen a$ is denoted as follows: :$n G := \gen {a^n}$ This notation is usually used in the context of $\struct {\Z, +}$, where $n \Z$ is (informally) understood as '''the set of integer multiples of $n$'''.


Quotient Group of Infinite Cyclic Group by Subgroup

Let $C_n$ be the cyclic group of order $n$. Then: :$C_n \cong \dfrac {\struct {\Z, +} } {\struct {n \Z, +} } = \dfrac \Z {n \Z}$ where: :$\Z$ is the additive group of integers :$n \Z$ is the additive group of integer multiples :$\Z / n \Z$ is the quotient group of $\Z$ by $n \Z$. Thus, every cyclic group is isomorphic to one of: :$\Z, \dfrac \Z \Z, \dfrac \Z {2 \Z}, \dfrac \Z {3 \Z}, \dfrac \Z {4 \Z}, \ldots$


Ring Operations on Coset Space of Ideal

Let $\struct {R, +, \circ}$ be a ring. Let $\powerset R$ be the power set of $R$. Let $J$ be an ideal of $R$. Let $X$ and $Y$ be cosets of $J$. Let $X +_\mathcal P Y$ be the sum of $X$ and $Y$, where $+_\mathcal P$ is the operation induced on $\powerset R$ by $+$. Similarly, let $X \circ_\mathcal P Y$ be the product of $X$ and $Y$, where $\circ_\mathcal P$ is the operation induced on $\powerset R$ by $\circ$. Then: * The sum $X +_\mathcal P Y$ in $\powerset R$ is also their sum in the quotient ring $R / J$. * The product $X \circ_\mathcal P Y$ in $\powerset R$ may be a proper subset of their product in $R / J$.


Property of Being an Ideal is not Transitive

Let $J_1$ be an ideal of a ring $R$. Let $J_2$ be an ideal of $J_1$. Then $J_2$ need not necessarily be an ideal of $R$.


Ideals Containing Ideal Form Lattice

Let $J$ be an ideal of a ring $R$. Let $\mathbb L_J$ be the set of all ideal of $R$ which contain $J$. Then the ordered set $\struct {\mathbb L_J, \subseteq}$ is a lattice.


Ideals Containing Ideal Isomorphic to Quotient Ring

Let $J$ be an ideal of a ring $R$. Let $\mathbb L_J$ be the set of all ideals of $R$ which contain $J$. Let the ordered set $\left({\mathbb L \left({R / J}\right), \subseteq}\right)$ be the set of all ideals of $R / J$. Let the mapping $\Phi_J: \left({\mathbb L_J, \subseteq}\right) \to \left({\mathbb L \left({R / J}\right), \subseteq}\right)$ be defined as: :$\forall a \in \mathbb L_J: \Phi_J \left({a}\right) = q_J \left({a}\right)$ where $q_J: a \to a / J$ is the quotient epimorphism from $a$ to $a / J$ from the definition of quotient ring. Then $\Phi_J$ is an isomorphism.


Ring of Integers is Principal Ideal Domain

The integers $\Z$ form a principal ideal domain.


Principal Ideals of Integers

Let $J$ be a non-zero ideal of $\Z$. Then $J = \ideal b$ where $b$ is the smallest strictly positive integer belonging to $J$.


Natural Numbers Set Equivalent to Ideals of Integers

Let the mapping $\psi: \N \to$ the set of all ideals of $\Z$ be defined as: :$\forall b \in \N: \psi \left({b}\right) = \left({b}\right)$ where $\left({b}\right)$ is the principal ideal of $\Z$ generated by $b$. Then $\psi$ is a bijection.


Canonical Epimorphism from Integers by Principal Ideal

Let $m$ be a strictly positive integer. Let $\left({m}\right)$ be the principal ideal of $\Z$ generated by $m$. The restriction to $\N_m$ of the canonical epimorphism $q_m$ from the ring $\left({\Z, +, \times}\right)$ onto $\left({\Z, +, \times}\right) / \left({m}\right)$ is an isomorphism from the ring $\left({\N_m, +_m, \times_m}\right)$ of integers modulo $m$ onto the quotient ring $\left({\Z, +, \times}\right) / \left({m}\right)$. In particular, $\left({\Z, +, \times}\right) / \left({m}\right)$ has $m$ elements.


Integer Divisor is Equivalent to Subset of Ideal

Let $\Z$ be the set of all integers. Let $\Z_{>0}$ be the set of strictly positive integers. Let $m \in \Z_{>0}$ and let $n \in \Z$. Let $\ideal m$ be the principal ideal of $\Z$ generated by $m$. Then: :$m \divides n \iff \ideal n \subseteq \ideal m$


Principal Ideals in Integral Domain

Let $\struct {D, +, \circ}$ be an integral domain. Let $U_D$ be the group of units of $D$. Let $\ideal x$ be the principal ideal of $D$ generated by $x$. Let $x, y \in \struct {D, +, \circ}$. Then:


Principal Ideal Domain is Unique Factorization Domain

Every principal ideal domain is a unique factorization domain.


Maximal Ideal iff Quotient Ring is Field

Let $\struct {R, +, \circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $J$ be an ideal of $R$. The following are equivalent: :$(1): \quad$ $J$ is a maximal ideal. :$(2): \quad$ The quotient ring $R / J$ is a field.


Prime Number iff Generates Principal Maximal Ideal

Let $\Z_{>0}$ be the set of strictly positive integers. Let $p \in \Z_{>0}$. Let $\ideal p$ be the principal ideal of $\Z$ generated by $p$.  Then $p$ is prime {{iff}} $\ideal p$ is a maximal ideal of $\Z$.


Integral Domain of Prime Order is Field

Let $\left({\Z_p, +_p, \times_p}\right)$ be the ring of integers modulo $p$. The following statements are equivalent: : $(1): \quad p$ is a prime. : $(2): \quad \left({\Z_p, +_p, \times_p}\right)$ is an integral domain. : $(3): \quad \left({\Z_p, +_p, \times_p}\right)$ is a field.


Quotient Ring of Integers and Zero

Let $\struct {\Z, +, \times}$ be the integral domain of integers. Let $\ideal 0$ be the principal ideal of $\struct {\Z, +, \times}$ generated by $0$. The quotient ring $\struct {\Z / \ideal 0, +, \times}$ is isomorphic to $\struct {\Z, +, \times}$.


Quotient Ring of Integers and Principal Ideal from Unity

Let $\left({\Z, +, \times}\right)$ be the integral domain of integers. Let $\left({1}\right)$ be the principal ideal of $\left({\Z, +, \times}\right)$ generated by $1$. The quotient ring $\left({\Z, +, \times}\right) / \left({1}\right)$ is isomorphic to the null ring.


Principal Ideal of Principal Ideal Domain is of Irreducible Element iff Maximal

Let $\struct {D, +, \circ}$ be a principal ideal domain. Let $\ideal p$ be the principal ideal of $D$ generated by $p$. Then $p$ is irreducible {{iff}} $\ideal p$ is a maximal ideal of $D$.


Subring Generated by Unity of Ring with Unity

Let $\left({R, +, \circ}\right)$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let the mapping $g: \Z \to R$ be defined as $\forall n \in \Z: g \left({n}\right) = n 1_R$, where $n 1_R$ the $n$th power of $1_R$. Let $\left({x}\right)$ be the principal ideal of $\left({R, +, \circ}\right)$ generated by $x$. Then $g$ is an epimorphism from $\Z$ onto the subring $S$ of $R$ generated by $1_R$. If $R$ has no proper zero divisors, then $g$ is the only nonzero homomorphism from $\Z$ into $R$. The kernel of $g$ is either: : $(1): \quad \left({0_R}\right)$, in which case $g$ is an isomorphism from $\Z$ onto $S$ or: : $(2): \quad \left({p}\right)$ for some prime $p$, in which case $S$ is isomorphic to the field $\Z_p$.


Null Ring iff Characteristic is One

The only ring whose characteristic is $1$ is the null ring.


Characteristic of Finite Ring with No Zero Divisors

Let $\struct {R, +, \circ}$ be a finite ring with unity with no proper zero divisors whose zero is $0_R$ and whose unity is $1_R$. Let $n \ne 0$ be the characteristic of $R$. Then: :$(1): \quad n$ must be a prime number :$(2): \quad n$ is the order of all non-zero elements in $\struct {R, +}$. It follows that $\struct {R, +} \cong C_n$, where $C_n$ is the cyclic group of order $n$.


Integral Domain with Characteristic Zero

In an integral domain with characteristic zero, every non-zero element has infinite order under ring addition.


Characteristic of Field is Zero or Prime

Let $F$ be a field. Then the characteristic of $F$ is either zero or a prime number.


Field of Characteristic Zero has Unique Prime Subfield

Let $F$ be a field, whose zero is $0_F$ and whose unity is $1_F$, with characteristic zero. Then there exists a unique $P \subseteq F$ such that: :$(1): \quad P$ is a subfield of $F$ :$(2): \quad P$ is isomorphic to the field of rational numbers $\struct {\Q, +, \times}$. That is, $P \cong \Q$ is a unique minimal subfield of $F$, and all other subfields of $F$ contain $P$. This field $P$ is called the prime subfield of $F$.


Field of Prime Characteristic has Unique Prime Subfield

Let $F$ be a field whose characteristic is $p$. Then there exists a unique $P \subseteq F$ such that: :$(1): \quad P$ is a subfield of $F$ :$(2): \quad P \cong \Z_p$. That is, $P \cong \Z_p$ is a unique minimal subfield of $F$, and all other subfields of $F$ contain $P$. This field $P$ is called the prime subfield of $F$.


Intersection of All Division Subrings is Prime Subfield

Let $\struct {K, +, \circ}$ be a division ring. Let $P$ be the intersection of the set of all division subrings of $K$. Then $P$ is the prime subfield of $K$.


Characteristic of Ordered Integral Domain is Zero

Let $\left({D, +, \circ}\right)$ be an ordered integral domain whose zero is $0_D$ and whose unity is $1_D$. Let $\operatorname{Char} \left({D}\right)$ be the characteristic of $D$. Then $\operatorname{Char} \left({D}\right) = 0$. Let $g: \Z \to D$ be the mapping defined as: :$\forall n \in \Z: g \left({n}\right) = n \cdot 1_D$ where $n \cdot 1_D$ is defined as the $n$th power of $1_D$. Then $g$ is the only monomorphism from the ordered ring $\Z$ onto the ordered ring $D$.


Monomorphism from Rational Numbers to Totally Ordered Field

Let $\struct {F, +, \circ, \le}$ be a totally ordered field. There is one and only one (ring) monomorphism from the totally ordered field $\Q$ onto $F$. Its image is the prime subfield of $F$.


Set of Polynomials over Integral Domain is Subring

Let $\struct {R, +, \circ}$ be a commutative ring. Let $\struct {D, +, \circ}$ be an integral subdomain of $R$. Then $\forall x \in R$, the set $D \sqbrk x$ of polynomials in $x$ over $D$ is a subring of $R$.


Polynomials Closed under Ring Product

Let $\left({R, +, \circ}\right)$ be a commutative ring. Let $\displaystyle f = \sum_{k \in Z} a_k \mathbf X^k$, $\displaystyle g =  \sum_{k \in Z} b_k \mathbf X^k$ be polynomials in the indeterminates $\left\{{X_j: j \in J}\right\}$ over $R$, where $Z$ is the set of all multiindices indexed by $\left\{{X_j: j \in J}\right\}$. Define the product :$\displaystyle f \otimes g = \sum_{k \in Z} c_k \mathbf X^k$ where :$\displaystyle c_k = \sum_{\substack{p + q = k \\ p, q \in Z}} a_p b_q$ Then $f \otimes g$ is a polynomial.


Ring of Polynomial Forms is Integral Domain

Let $\struct {R, +, \circ}$ be a commutative ring with unity. Let $\struct {D, +, \circ}$ be an integral subdomain of $R$. Let $X \in R$ be transcendental over $D$. Let $D \sqbrk X$ be the ring of polynomials in $X$ over $D$. Then $D \sqbrk X$ is an integral domain.


Injection is Bijection iff Inverse is Injection

Let $\phi: S \to T$ be an injection. Then $\phi$ is a bijection {{iff}} its inverse $\phi^{-1}$ is also an injection.


Epimorphism from Polynomial Forms to Polynomial Functions

Let $D$ be an integral domain. Let $D \sqbrk X$ be the ring of polynomial forms in $X$ over $D$. Let $\map P D$ be the ring of polynomial functions over $D$. The mapping $\kappa: D \sqbrk X \to \map P D$ given by: :$\displaystyle \map \kappa {\sum_{k \mathop = 0}^n {a_k \circ X^k} } = f$ where $\displaystyle f = \sum_{k \mathop = 0}^n {a_k \circ x^k}, x \in D$ is a ring epimorphism.


Division Theorem for Polynomial Forms over Field

Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $X$ be transcendental over $F$. Let $F \sqbrk X$ be the ring of polynomials in $X$ over $F$. Let $d$ be an element of $F \sqbrk X$ of degree $n \ge 1$. Then $\forall f \in F \sqbrk X: \exists q, r \in F \sqbrk X: f = q \circ d + r$ such that either: :$(1): \quad r = 0_F$ or: :$(2): \quad r \ne 0_F$ and $r$ has degree that is less than $n$.


Polynomial Forms over Field form Principal Ideal Domain

Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $X$ be transcendental over $F$. Let $F \sqbrk X$ be the ring of polynomials in $X$ over $F$. Then $F \sqbrk X$ is a principal ideal domain.


Equal Consecutive Prime Number Gaps are Multiples of Six

If you list the gaps between consecutive primes greater than $5$: :$2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, \ldots$ you will notice that consecutive gaps that are equal are of the form $6 x$. This is ''always'' the case. {{OEIS|A001223}}


Standard Discrete Metric is Metric

The standard discrete metric is a metric.


Derivative of Constant

Let $\map {f_c} x$ be the constant function on $\R$, where $c \in \R$. Then: :$\map {f_c'} x = 0$


Derivative of Identity Function

Let $X$ be either set of either the real numbers $\R$ or the complex numbers $\C$. Let $I_X: X \to X$ be the identity function. Then: :$\map {I_X'} x = 1$ where $\map {I_X'} x$ denotes the derivative of $I_X$ {{WRT|Differentiation}} $x$. This can be presented for each of $\R$ and $\C$:


Product Rule for Derivatives

Let $\map f x, \map j x, \map k x$ be real functions defined on the open interval $I$. Let $\xi \in I$ be a point in $I$ at which both $j$ and $k$ are differentiable. Let $\map f x = \map j x \map k x$. Then: :$\map {f'} \xi = \map j \xi \map {k'} \xi + \map {j'} \xi \map k \xi$ It follows from the definition of derivative that if $j$ and $k$ are both differentiable on the interval $I$, then: :$\forall x \in I: \map {f'} x = \map j x \map {k'} x + \map {j'} x \map k x$ Using Leibniz's notation for derivatives, this can be written as: :$\map {\dfrac \d {\d x} } {y \, z} = y \dfrac {\d z} {\d x} + \dfrac {\d y} {\d x} z$ where $y$ and $z$ represent functions of $x$.


Derivative of Composite Function

Let $f, g, h$ be continuous real functions such that: :$\forall x \in \R: \map h x = \map {f \circ g} x = \map f {\map g x}$ Then: :$\map {h'} x = \map {f'} {\map g x} \map {g'} x$ where $h'$ denotes the derivative of $h$. Using the $D_x$ notation: :$\map {D_x} {\map f {\map g x} } = \map {D_{\map g x} } {\map f {\map g x} } \, \map {D_x} {\map g x}$ This is often informally referred to as the '''chain rule (for differentiation)'''.


Derivative of Inverse Function

Let $I = \closedint a b$ and $J = \closedint c d$ be closed real intervals. Let $I^o = \openint a b$ and $J^o = \openint c d$ be the corresponding open real intervals. Let $f: I \to J$ be a real function which is continuous on $I$ and differentiable on $I^o$ such that $J = f \sqbrk I$. Let either: :$\forall x \in I^o: D \map f x > 0$ or: :$\forall x \in I^o: D \map f x < 0$ Then: :$f^{-1}: J \to I$ exists and is continuous on $J$ :$f^{-1}$ is differentiable on $J^o$ :$\forall y \in J^o: D \map {f^{-1} } y = \dfrac 1 {D \map f x}$


Upper Sum Never Smaller than Lower Sum

Let $\closedint a b$ be a closed interval of the set $\R$ of real numbers. Let $P = \set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ be a finite subdivision of $\closedint a b$. Let $f: \R \to \R$ be a real function. Let $f$ be bounded on $\closedint a b$. Let $\map L P$ be the lower sum of $\map f x$ on $\closedint a b$ belonging to the subdivision $P$. Let $\map U P$ be the upper sum of $\map f x$ on $\closedint a b$ belonging to the subdivision $P$. Then $\map L P \le \map U P$.


Dedekind's Theorem

Let $\tuple {L, R}$ be a Dedekind cut of the set of real numbers $\R$. Then there exists a unique real number which is a producer of $\tuple {L, R}$.


Complex Addition is Closed

The set of complex numbers $\C$ is closed under addition: :$\forall z, w \in \C: z + w \in \C$


Complex Addition is Associative

The operation of addition on the set of complex numbers $\C$ is associative: :$\forall z_1, z_2, z_3 \in \C: z_1 + \paren {z_2 + z_3} = \paren {z_1 + z_2} + z_3$


Complex Addition is Commutative

The operation of addition on the set of complex numbers is commutative: :$\forall z, w \in \C: z + w = w + z$


Integers are Countably Infinite

The set $\Z$ of integers is countably infinite.


Permutation of Determinant Indices

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$ over a field. Let $\lambda: \N_{> 0} \to \N_{> 0}$ be any fixed permutation on $\N_{> 0}$. Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$. Let $\struct {S_n, \circ}$ be the symmetric group of $n$ letters.  Then: :$\displaystyle \map \det {\mathbf A} = \sum_{\mu \mathop \in S_n} \paren {\map \sgn \mu \map \sgn \lambda \prod_{k \mathop = 1}^n a_{\map \lambda k, \map \mu k} }$ :$\displaystyle \map \det {\mathbf A} = \sum_{\mu \mathop \in S_n} \paren {\map \sgn \mu \map \sgn \lambda \prod_{k \mathop = 1}^n a_{\map \mu k, \map \lambda k} }$ where: :the summation $\displaystyle \sum_{\mu \mathop \in S_n}$ goes over all the $n!$ permutations of $\set {1, 2, \ldots, n}$ :$\map \sgn \mu$ is the sign of the permutation $\mu$.


Determinant of Transpose

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$. Let $\mathbf A^\intercal$ be the transpose of $\mathbf A$. Then: :$\map \det {\mathbf A} = \map \det {\mathbf A^\intercal}$


Determinant with Rows Transposed

If two rows of a matrix with determinant $D$ are transposed, its determinant becomes $-D$.


Square Matrix with Duplicate Rows has Zero Determinant

If two rows of a square matrix over a commutative ring $\struct {R, +, \circ}$ are the same, then its determinant is zero.


Determinant with Row Multiplied by Constant

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$. Let $\mathbf B$ be the matrix resulting from one row of $\mathbf A$ having been multiplied by a constant $c$. Then: :$\map \det {\mathbf B} = c \map \det {\mathbf A}$ That is, multiplying one row of a square matrix by a constant multiplies its determinant by that constant.


Determinant as Sum of Determinants

Let $\begin{vmatrix}   a_{11} & \cdots & a_{1s} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\   a_{r1} & \cdots & a_{rs} & \cdots & a_{rn} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\   a_{n1} & \cdots & a_{ns} & \cdots & a_{nn} \end{vmatrix}$ be a determinant. Then $\begin{vmatrix}   a_{11} & \cdots & a_{1s} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\   a_{r1} + a'_{r1} & \cdots & a_{rs} + a'_{rs} & \cdots & a_{rn} + a'_{rn} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\   a_{n1} & \cdots & a_{ns} & \cdots & a_{nn} \end{vmatrix} = \begin{vmatrix}   a_{11} & \cdots & a_{1s} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\   a_{r1} & \cdots & a_{rs} & \cdots & a_{rn} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\   a_{n1} & \cdots & a_{ns} & \cdots & a_{nn} \end{vmatrix} + \begin{vmatrix}   a_{11} & \cdots & a_{1s} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\   a'_{r1} & \cdots & a'_{rs} & \cdots & a'_{rn} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\   a_{n1} & \cdots & a_{ns} & \cdots & a_{nn} \end{vmatrix}$. Similarly: Then $\begin{vmatrix}   a_{11} & \cdots & a_{1s} + a'_{1s} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\   a_{r1} & \cdots & a_{rs} + a'_{rs} & \cdots & a_{rn} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\   a_{n1} & \cdots & a_{ns} + a'_{ns} & \cdots & a_{nn} \end{vmatrix} = \begin{vmatrix}   a_{11} & \cdots & a_{1s} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\   a_{r1} & \cdots & a_{rs} & \cdots & a_{rn} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\   a_{n1} & \cdots & a_{ns} & \cdots & a_{nn} \end{vmatrix} + \begin{vmatrix}   a_{11} & \cdots & a'_{1s} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\   a_{r1} & \cdots & a'_{rs} & \cdots & a_{rn} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\   a_{n1} & \cdots & a'_{ns} & \cdots & a_{nn} \end{vmatrix}$.


Multiple of Row Added to Row of Determinant

Let $\mathbf A = \begin {bmatrix} a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\  \vdots &  \vdots & \ddots &  \vdots \\ a_{r 1} & a_{r 2} & \cdots & a_{r n} \\  \vdots &  \vdots & \ddots &  \vdots \\ a_{s 1} & a_{s 2} & \cdots & a_{s n} \\  \vdots &  \vdots & \ddots &  \vdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n} \\ \end {bmatrix}$ be a square matrix of order $n$. Let $\map \det {\mathbf A}$ denote the determinant of $\mathbf A$. Let $\mathbf B = \begin{bmatrix} a_{1 1}             & a_{1 2}             & \cdots &             a_{1 n} \\  \vdots             & \vdots              & \ddots &              \vdots \\ a_{r 1} + k a_{s 1} & a_{r 2} + k a_{s 2} & \cdots & a_{r n} + k a_{s n} \\  \vdots             & \vdots              & \ddots &              \vdots \\ a_{s 1}             & a_{s 2}             & \cdots &             a_{s n} \\  \vdots             & \vdots              & \ddots &              \vdots \\ a_{n 1}             & a_{n 2}             & \cdots &             a_{n n} \\ \end{bmatrix}$. Then $\map \det {\mathbf B} = \map \det {\mathbf A}$. That is, the value of a determinant remains unchanged if a constant multiple of any row is added to any other row.


Determinant of Matrix Product

Let $\mathbf A = \sqbrk a_n$ and $\mathbf B = \sqbrk b_n$ be a square matrices of order $n$. Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$. Let $\mathbf A \mathbf B$ be the (conventional) matrix product of $\mathbf A$ and $\mathbf B$. Then: :$\map \det {\mathbf A \mathbf B} = \map \det {\mathbf A} \map \det {\mathbf B}$ That is, the determinant of the product is equal to the product of the determinants.


Expansion Theorem for Determinants

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. Let $D = \map \det {\mathbf A}$ be the determinant of $\mathbf A$: :$\displaystyle \map \det {\mathbf A} := \sum_{\lambda} \paren {\map \sgn \lambda \prod_{k \mathop = 1}^n a_{k \map \lambda k} } = \sum_\lambda \map \sgn \lambda a_{1 \map \lambda 1} a_{2 \map \lambda 2} \cdots a_{n \map \lambda n}$ where: :the summation $\displaystyle \sum_\lambda$ goes over all the $n!$ permutations of $\set {1, 2, \ldots, n}$ :$\map \sgn \lambda$ is the sign of the permutation $\lambda$. Let $a_{p q}$ be an element of $\mathbf A$. Let $A_{p q}$ be the cofactor of $a_{p q}$ in $D$. Then: :$(1): \quad \displaystyle \forall r \in \closedint 1 n: D = \sum_{k \mathop = 1}^n a_{r k} A_{r k}$ :$(2): \quad \displaystyle \forall r \in \closedint 1 n: D = \sum_{k \mathop = 1}^n a_{k r} A_{k r}$ Thus the value of a determinant can be found either by: :multiplying all the elements in a row by their cofactors and adding up the products or: :multiplying all the elements in a column by their cofactors and adding up the products. The identity: :$\displaystyle D = \sum_{k \mathop = 1}^n a_{r k} A_{r k}$ is known as the '''expansion of $D$ in terms of row $r$''', while: :$\displaystyle D = \sum_{k \mathop = 1}^n a_{k r} A_{k r}$ is known as the '''expansion of $D$ in terms of column $r$'''.


Determinant with Unit Element in Otherwise Zero Row

Let $D$ be the determinant: :$D = \begin {vmatrix}       1 &       0 & \cdots &       0 \\ b_{2 1} & b_{2 2} & \cdots & b_{2 n} \\  \vdots &  \vdots & \ddots &  \vdots \\ b_{n 1} & b_{n 2} & \cdots & b_{n n} \end {vmatrix}$ Then: :$D = \begin {vmatrix} b_{2 2} & \cdots & b_{2 n} \\  \vdots & \ddots &  \vdots \\ b_{n 2} & \cdots & b_{n n} \end {vmatrix}$


Vandermonde Determinant

The '''Vandermonde determinant of order $n$''' is the determinant defined as follows: :$V_n = \begin {vmatrix} 1 & x_1 & x_1^2 & \cdots & x_1^{n - 2} & x_1^{n - 1} \\ 1 & x_2 & x_2^2 & \cdots & x_2^{n - 2} & x_2^{n - 1} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & x_n & x_n^2 & \cdots & x_n^{n - 2} & x_n^{n - 1} \end {vmatrix}$ Its value is given by: :$\displaystyle V_n = \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_j - x_i}$


Equality of Polynomials

$f$ and $g$ are equal as polynomials {{iff}} $f$ and $g$ are equal as functions. Thus we can say $f = g$ without ambiguity as to what it means. {{explain|In the exposition, the term was "equal as forms", but it has now morphed into "equal as polynomials". Needs to be resolved.}}


Value of Adjugate of Determinant

Let $D$ be the determinant of order $n$. Let $D^*$ be the adjugate of $D$. Then $D^* = D^{n-1}$.


Determinant of Diagonal Matrix

Let $\mathbf A = \begin{bmatrix} a_{11} & 0 & \cdots & 0 \\ 0 & a_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_{nn} \\ \end{bmatrix}$ be a diagonal matrix. Then the determinant of $\mathbf A$ is the product of the elements of $\mathbf A$. That is: :$\ds \map \det {\mathbf A} = \prod_{i \mathop = 1}^n a_{ii}$


Matrix is Row Equivalent to Reduced Echelon Matrix

Let $\mathbf A = \sqbrk a_{m n}$ be a matrix of order $m \times n$ over a field $F$. Then $A$ is row equivalent to a reduced echelon matrix of order $m \times n$.


Square Matrix is Row Equivalent to Triangular Matrix

Let $\mathbf A = \left[{a}\right]_n$ be a square matrix of order $n$ over a commutative ring $R$. Then $\mathbf A$ can be converted to an upper or lower triangular matrix by elementary row operations.


Product of Triangular Matrices

Let $\mathbf A = \sqbrk a_n, \mathbf B = \sqbrk b_n$ be upper triangular matrices of order $n$. Let $\mathbf C = \mathbf A \mathbf B$. Then :$(1): \quad$ the diagonal elements of $\mathbf C$ are given by: ::::$\forall j \in \closedint 1 n: c_{j j} = a_{j j} b_{j j}$ :::That is, the diagonal elements of $\mathbf C$ are those of the factor matrices multiplied together. :$(2): \quad$ The matrix $\mathbf C$ is itself upper triangular. The same applies if both $\mathbf A$ and $\mathbf B$ are lower triangular matrices.


Effect of Elementary Row Operations on Determinant

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. Let $\map \det {\mathbf A}$ denote the determinant of $\mathbf A$. Take the elementary row operations: {{begin-axiom}} {{axiom | n = \text {ERO} 1         | t = For some $\lambda$, multiply row $i$ by $\lambda$         | m = r_i \to \lambda r_i }} {{axiom | n = \text {ERO} 2         | t = For some $\lambda$, add $\lambda$ times row $j$ to row $i$         | m = r_i \to r_i + \lambda r_j }} {{axiom | n = \text {ERO} 3         | t = Exchange rows $i$ and $j$         | m = r_i \leftrightarrow r_j }} {{end-axiom}} Applying $\text {ERO} 1$ has the effect of multiplying $\map \det {\mathbf A}$ by $\lambda$. Applying $\text {ERO} 2$ has no effect on $\map \det {\mathbf A}$. Applying $\text {ERO} 3$ has the effect of multiplying $\map \det {\mathbf A}$ by $-1$.


Modulus in Terms of Conjugate

Let $z = a + i b$ be a complex number. Let $\cmod z$ be the modulus of $z$. Let $\overline z$ be the conjugate of $z$. Then: :$\cmod z^2 = z \overline z$


Existence and Uniqueness of Positive Root of Positive Real Number

Let $x \in \R$ be a real number such that $x \ge 0$. Let $n \in \Z$ be an integer such that $n \ne 0$. Then there always exists a unique $y \in \R: \paren {y \ge 0} \land \paren {y^n = x}$. Hence the justification for the terminology '''the positive $n$th root of $x$''' and the notation $x^{1/n}$.


Triangle Inequality

=== Geometry === {{:Triangle Inequality/Geometry}} === Real Numbers === {{:Triangle Inequality/Real Numbers}} === Complex Numbers === {{:Triangle Inequality/Complex Numbers}}


Even Power is Non-Negative

Let $x \in \R$ be a real number. Let $n \in \Z$ be an even integer. Then $x^n \ge 0$. That is, all even powers are positive.


Sign of Odd Power

Let $x \in \R$ be a real number. Let $n \in \Z$ be an odd integer. Then: :$x^n = 0 \iff x = 0$ :$x^n > 0 \iff x > 0$ :$x^n < 0 \iff x < 0$ That is, the sign of an odd power matches the number it is a power of.


Product of Absolute Values on Ordered Integral Domain

Let $\struct {D, +, \times, \le}$ be an ordered integral domain whose zero is denoted by $0_D$. For all $a \in D$, let $\size a$ denote the absolute value of $a$. Then: :$\size a \times \size b = \size {a \times b}$


Negative of Absolute Value

Let $x \in \R$ be a real number. Let $\size x$ denote the absolute value of $x$. Then: :$-\size x \le x \le \size x$


Order of Squares in Ordered Ring

Let $\struct {R, +, \circ, \le}$ be an ordered ring whose zero is $0_R$ and whose unity is $1_R$. Let $x, y \in \struct {R, +, \circ, \le}$ such that $0_R \le x, y$. Then: :$x \le y \implies x \circ x \le y \circ y$ When $R$ is one of the standard sets of numbers, that is $\Z, \Q, \R$, then this translates into: :If $x, y$ are positive then $x \le y \implies x^2 \le y^2$.


Continuum Property

Let $S \subset \R$ be a non-empty subset of the set of real numbers such that $S$ is bounded above. Then $S$ admits a supremum in $\R$. This is known as the '''least upper bound property''' of the real numbers. Similarly, let $S \subset \R$ be a non-empty subset of the set of real numbers such that $S$ is bounded below. Then $S$ admits an infimum in $\R$. This is sometimes called the '''greatest lower bound property''' of the real numbers. The two properties taken together are called the '''continuum property of $\R$'''. This can also be stated as: :The set $\R$ of real numbers is Dedekind complete.


Complex Multiplication is Closed

The set of complex numbers $\C$ is closed under multiplication: :$\forall z, w \in \C: z \times w \in \C$


Complex Multiplication is Associative

The operation of multiplication on the set of complex numbers $\C$ is associative: :$\forall z_1, z_2, z_3 \in \C: z_1 \paren {z_2 z_3} = \paren {z_1 z_2} z_3$


Arithmetic Mean is Never Less than Harmonic Mean

Let $x_1, x_2, \ldots, x_n \in \R_{> 0}$ be strictly positive real numbers. Let $A_n $ be the arithmetic mean of $x_1, x_2, \ldots, x_n$. Let $H_n$ be the harmonic mean of $x_1, x_2, \ldots, x_n$. Then $A_n \ge H_n$.


Multiple of Supremum

Let $S \subseteq \R: S \ne \varnothing$ be a non-empty subset of the set of real numbers $\R$. Let $S$ be bounded above. Let $z \in \R: z > 0$ be a positive real number. Then: :$\displaystyle \map {\sup_{x \mathop \in S} } {z x} = z \map {\sup_{x \mathop \in S} } x$


Cauchy's Mean Theorem

Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers which are all positive. Let $A_n$ be the arithmetic mean of $x_1, x_2, \ldots, x_n$. Let $G_n$ be the geometric mean of $x_1, x_2, \ldots, x_n$. Then: :$A_n \ge G_n$ with equality holding {{iff}}: :$\forall i, j \in \set {1, 2, \ldots, n}: x_i = x_j$ That is, {{iff}} all terms are equal.


Distance on Real Numbers is Metric

Let $x, y \in \R$ be real numbers. Let $\map d {x, y}$ be the distance between $x$ and $y$: :$\map d {x, y} = \size {x - y}$ Then $\map d {x, y}$ is a metric on $\R$. Thus it follows that $\tuple {\R, d}$ is a metric space.


Combination Theorem for Sequences/Real

Let $\sequence {x_n}$ and $\sequence {y_n}$ be sequences in $\R$. Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent to the following limits: :$\displaystyle \lim_{n \mathop \to \infty} x_n = l$ :$\displaystyle \lim_{n \mathop \to \infty} y_n = m$ Let $\lambda, \mu \in \R$. Then the following results hold: === Sum Rule === {{:Combination Theorem for Sequences/Real/Sum Rule}} === Difference Rule === {{:Combination Theorem for Sequences/Real/Difference Rule}} === Multiple Rule === {{:Combination Theorem for Sequences/Real/Multiple Rule}} === Combined Sum Rule === {{:Combination Theorem for Sequences/Real/Combined Sum Rule}} === Product Rule === {{:Combination Theorem for Sequences/Real/Product Rule}} === Quotient Rule === {{:Combination Theorem for Sequences/Real/Quotient Rule}}


Convergent Sequence in Metric Space is Bounded

Let $M = \struct {A, d}$ be a metric space. Let $\sequence {x_n}$ be a sequence in $M$ which is convergent, and so $x_n \to l$ as $n \to \infty$. Then $\sequence {x_n}$ is bounded.


Convergent Sequence Minus Limit

Let $X$ be  one of the standard number fields $\Q, \R, \C$. Let $\left \langle {x_n} \right \rangle$ be a sequence in $X$ which converges to $l$. That is: : $\displaystyle \lim_{n \mathop \to \infty} x_n = l$ Then: : $\displaystyle \lim_{n \mathop \to \infty} \left|{x_n - l}\right| = 0$


Monotone Convergence Theorem (Real Analysis)

Every bounded monotone sequence is convergent.


One Plus Reciprocal to the Nth

Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = \paren {1 + \dfrac 1 n}^n$. Then $\sequence {x_n}$ converges to a limit as $n$ increases without bound.


Between two Real Numbers exists Rational Number

Let $a, b \in \R$ be real numbers such that $a < b$. Then: : $\exists r \in \Q: a < r < b$


Power over Factorial

Let $x \in \R: x > 0$ be a positive real number. Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = \dfrac {x^n} {n!}$. Then $\sequence {x_n}$ converges to zero.


Sequence of Powers of Reciprocals is Null Sequence

Let $r \in \Q_{>0}$ be a strictly positive rational number. Let $\sequence {x_n}$ be the sequence in $\R$ defined as: : $x_n = \dfrac 1 {n^r}$ Then $\sequence {x_n}$ is a null sequence.


Euler's Number: Limit of Sequence implies Limit of Series

Let Euler's number $e$ be defined as: :$\displaystyle e := \lim_{n \to \infty} \left({1 + \frac 1 n}\right)^n$ Then: :$\displaystyle e = \sum_{k \mathop \ge 0} \frac 1 {k!}$ That is: :$\displaystyle e = \frac 1 {0!} + \frac 1 {1!} + \frac 1 {2!} + \frac 1 {3!} + \frac 1 {4!} \cdots$


Degree of Field Extensions is Multiplicative

Let $E, K$ and $F$ be fields. Let $E / K$ and $K / F$ be finite field extensions. Then $E / F$ is a finite field extension, and: :$\index E F = \index E K \index K F$ where $\index E F$ denotes the degree of $E / F$


Convergent Sequence in Metric Space has Unique Limit

Let $\left({X, d}\right)$ be a metric space. Let $\left \langle {x_n} \right \rangle$ be a sequence in $\left({X, d}\right)$. Then $\left \langle {x_n} \right \rangle$ can have at most one limit.


Lower and Upper Bounds for Sequences

Let $\sequence {x_n}$ be a sequence in $\R$. Let $x_n \to l$ as $n \to \infty$. Then: :$(1): \quad \forall n \in \N: x_n \ge a \implies l \ge a$ :$(2): \quad \forall n \in \N: x_n \le b \implies l \le b$


Ring Homomorphism from Division Ring is Monomorphism or Zero Homomorphism

Let $\struct {R, +_R, \circ}$ and $\struct {S, +_S, *}$ be rings whose zeros are $0_R$ and $0_S$ respectively. Let $\phi: R \to S$ be a ring homomorphism. If $R$ is a division ring, then either: : $(1): \quad \phi$ is a monomorphism (that is, $\phi$ is injective) : $(2): \quad \phi$ is the zero homomorphism (that is, $\forall a \in R: \map \phi a = 0_S$).


Kernel of Ring Epimorphism is Ideal

Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ be a ring epimorphism. Then: :The kernel of $\phi$ is an ideal of $R_1$. :There is a unique ring isomorphism $g: R_1 / K \to R_2$ such that: ::$g \circ q_K = \phi$ :$\phi$ is a ring isomorphism {{iff}} $K = \set {0_{R_1} }$.


Unbounded Monotone Sequence Diverges to Infinity

Let $\sequence {x_n}$ be a sequence in $\R$. Let $\sequence {x_n}$ be monotone, that is either increasing or decreasing.


Reciprocal of Null Sequence

Let $\sequence {x_n}$ be a sequence in $\R$. Let $\forall n \in \N: x_n > 0$. Then: :$x_n \to 0$ as $n \to \infty$ {{iff}} $\size {\dfrac 1 {x_n} } \to \infty$ as $n \to \infty$


Modulus of Limit

Let $X$ be one of the standard number fields $\Q, \R, \C$. Let $\sequence {x_n}$ be a sequence in $X$. Let $\sequence {x_n}$ be convergent to the limit $l$. That is, let $\displaystyle \lim_{n \mathop \to \infty} x_n = l$. Then :$\displaystyle \lim_{n \mathop \to \infty} \cmod {x_n} = \cmod l$ where $\cmod {x_n}$ is the modulus of $x_n$.


Strictly Increasing Sequence of Natural Numbers

Let $\N_{>0}$ be the set of natural numbers without zero: : $\N_{>0} = \left\{{1, 2, 3, \ldots}\right\}$ Let $\left \langle {n_r} \right \rangle$ be strictly increasing sequence in $\N_{>0}$. Then: : $\forall r \in \N_{>0}: n_r \ge r$


Limit of Subsequence equals Limit of Sequence

Let $T = \struct {S, \tau}$ be a topological space. Let $\sequence {x_n}$ be a sequence in $T$. Let $l \in S$ such that: :$\displaystyle \lim_{n \mathop \to \infty} x_n = l$ Let $\sequence {x_{n_r} }$ be a subsequence of $\sequence {x_n}$. Then: :$\displaystyle \lim_{r \mathop \to \infty} x_{n_r} = l$ That is, the limit of a convergent sequence in a topological space equals the limit of any subsequence of it.


Root of Number Greater than One

Let $x \in \R$ be a real number. Let $n \in \N^*$ be a natural number such that $n > 0$. Then $x \ge 1 \implies x^{1/n} \ge 1$ where $x^{1/n}$ is the $n$th root of $x$.


Limit of Root of Positive Real Number

Let $x \in \R: x > 0$ be a real number. Let $\sequence {x_n}$ be the sequence in $\R$ defined as: :$x_n = x^{1 / n}$ Then $x_n \to 1$ as $n \to \infty$.


Hero's Method

Let $a \in \R$ be a real number such that $a > 0$. Let $x_1 \in \R$ be a real number such that $x_1 > 0$. Let $\sequence {x_n}$ be the sequence in $\R$ defined recursively by: :$\forall n \in \N_{>0}: x_{n + 1} = \dfrac {x_n + \dfrac a {x_n} } 2$ Then $x_n \to \sqrt a$ as $n \to \infty$.


Limit of Integer to Reciprocal Power

Let $\sequence {x_n}$ be the real sequence defined as $x_n = n^{1/n}$, using exponentiation. Then $\sequence {x_n}$ converges with a limit of $1$.


Difference Between Adjacent Square Roots Converges

Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = \sqrt {n + 1} - \sqrt n$. Then $\sequence {x_n}$ converges to a zero limit.


Peak Point Lemma

Let $\sequence {x_n}$ be a sequence in $\R$ which is infinite. Then $\sequence {x_n}$ has an infinite subsequence which is monotone.


Bolzano-Weierstrass Theorem

Every bounded sequence of real numbers has a convergent subsequence.


Existence of Maximum and Minimum of Bounded Sequence

Let $\sequence {x_n}$ be a bounded sequence in $\R$ (which may or may not be convergent). Let $L$ be the set of all real numbers which are the limit of some subsequence of $\sequence {x_n}$. Then $L$ has both a maximum and a minimum.


Interval Defined by Absolute Value

Let $\xi, \delta \in \R$ be real numbers. Let $\delta > 0$. Then:


Difference of Two Powers

Let $\mathbb F$ denote one of the standard number systems, that is $\Z$, $\Q$, $\R$ and $\C$. Let $n \in \N$ such that $n \ge 2$. Then for all $a, b \in \mathbb F$: {{begin-eqn}} {{eqn | l = a^n - b^n       | r = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j       | c =  }} {{eqn | r = \paren {a - b} \paren {a^{n - 1} + a^{n - 2} b + a^{n - 3} b^2 + \dotsb + a b^{n - 2} + b^{n - 1} }       | c =  }} {{end-eqn}}


Polynomial Factor Theorem

Let $P \paren x$ be a polynomial in $x$ over a field $K$ of degree $n$. Then: :$\xi \in K: P \paren \xi = 0 \iff \map P x = \paren {x - \xi} \map Q x$ where $Q$ is a polynomial of degree $n - 1$. Hence, if $\xi_1, \xi_2, \ldots, \xi_n \in K$ such that all are different, and $\map P {\xi_1} = \map P {\xi_2} = \dotsb = \map P {\xi_n} = 0$, then: :$\displaystyle \map P x = k \prod_{j \mathop = 1}^n \paren {x - \xi_j}$ where $k \in K$.


Telescoping Series/Example 1

Let $\left \langle {b_n} \right \rangle$ be a sequence in $\R$. Let $\left \langle {a_n} \right \rangle$ be a sequence whose terms are defined as: :$a_k = b_k - b_{k + 1}$ Then: :$\displaystyle \sum_{k \mathop = 1}^n a_k = b_1 - b_{n + 1}$


Terms in Convergent Series Converge to Zero

Let $\sequence {a_n}$ be a sequence in any of the standard number fields $\Q$, $\R$, or $\C$. Suppose that the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ converges in any of the standard number fields $\Q$, $\R$, or $\C$. Then: :$\displaystyle \lim_{n \mathop \to \infty} a_n = 0$ {{expand|Expand (on a different page) to Banach spaces}}


Linear Combination of Convergent Series

Let $\sequence {a_n}_{n \mathop \ge 1}$ and $\sequence {b_n}_{n \mathop \ge 1}$ be sequences of real numbers. Let the two series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ and $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ converge to $\alpha$ and $\beta$ respectively. Let $\lambda, \mu \in \R$ be real numbers. Then the series $\displaystyle \sum_{n \mathop = 1}^\infty \paren {\lambda a_n + \mu b_n}$ converges to $\lambda \alpha + \mu \beta$.


Abel's Theorem

Let $\displaystyle \sum_{k \mathop = 0}^\infty a_k$ be a convergent series in $\R$. Then: :$\displaystyle \lim_{x \mathop \to 1^-} \paren {\sum_{k \mathop = 0}^\infty a_k x^k} = \sum_{k \mathop = 0}^\infty a_k$ where $\displaystyle \lim_{x \mathop \to 1^-}$ denotes the limit from the left.


Alternating Series Test

Let $\sequence {a_n}_{N \mathop \ge 0}$ be a decreasing sequence of positive terms in $\R$ which converges with a limit of zero. That is, let $\forall n \in \N: a_n \ge 0, a_{n + 1} \le a_n, a_n \to 0$ as $n \to \infty$ Then the series: :$\displaystyle \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} a_n = a_1 - a_2 + a_3 - a_4 + \dotsb$  converges.


Euler's Identity

:$e^{i \pi} + 1 = 0$


Intermediate Value Theorem

Let $f: S \to \R$ be a real function on some subset $S$ of $\R$. Let $I \subseteq S$ be a real interval. Let $f: I \to \R$ be continuous on $I$. Let $a, b \in I$. Let $k \in \R$ lie between $\map f a$ and $\map f b$. That is, either: :$\map f a < k < \map f b$ or: :$\map f b < k < \map f a$ Then $\exists c \in \openint a b$ such that $\map f c = k$.


Tail of Convergent Series tends to Zero

Let $\sequence {a_n}_{n \mathop \ge 1}$ be a sequence of real numbers. Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a convergent series. Let $N \in \N_{\ge 1}$ be a natural number. Let $\displaystyle \sum_{n \mathop = N}^\infty a_n$ be the tail of the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$. Then: :$\displaystyle \sum_{n \mathop = N}^\infty a_n$ is convergent :$\displaystyle \sum_{n \mathop = N}^\infty a_n \to 0$ as $N \to \infty$. That is, the tail of a convergent series tends to zero.


Comparison Test

Let $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ be a convergent series of positive real numbers. Let $\sequence {a_n}$ be a sequence $\R$ or sequence in $\C$. Let $\forall n \in \N_{>0}: \cmod {a_n} \le b_n$. Then the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ converges absolutely.


Ratio Test

Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a series of real numbers in $\R$, or a series of complex numbers in $\C$. Let the sequence $\sequence {a_n}$ satisfy: :$\displaystyle \lim_{n \mathop \to \infty} \size {\frac {a_{n + 1} } {a_n} } = l$ :If $l > 1 $, then $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ diverges. :If $l < 1 $, then $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ converges absolutely.


Limit of Subsequence of Bounded Sequence

Let $\sequence {x_n}$ be a sequence in $\R$. Let $\sequence {x_n}$ be bounded. Let $b \in \R$ be a real number. Suppose that $\forall N: \exists n > N: x_n \ge b$. Then $\sequence {x_n}$ has a subsequence which converges to a limit $l \ge b$.


Terms of Bounded Sequence Within Bounds

Let $\sequence {x_n}$ be a sequence in $\R$. Let $\sequence {x_n}$ be bounded. Let the limit superior of $\sequence {x_n}$ be $\overline l$. Let the limit inferior of $\sequence {x_n}$ be $\underline l$. Then: :$\forall \epsilon > 0: \exists N: \forall n > N: x_n < \overline l + \epsilon$ :$\forall \epsilon > 0: \exists N: \forall n > N: x_n > \underline l - \epsilon$


Convergence of Limsup and Liminf

Let $\sequence {x_n}$ be a sequence in $\R$. Let the limit superior of $\sequence {x_n}$ be $\overline l$. Let the limit inferior of $\sequence {x_n}$ be $\underline l$. Then $\left \langle {x_n} \right \rangle$ converges to a limit $l$ {{iff}} $\overline l = \underline l = l$. Hence a bounded real sequence converges {{iff}} all its convergent subsequences have the same limit.


Limsup and Liminf are Limits of Bounds

Let $\sequence {x_n}$ be a sequence in $\R$. Let $\sequence {x_n}$ be bounded. Let $\displaystyle \overline l = \limsup_{n \mathop \to \infty} x_n$ be the limit superior and $\displaystyle \liminf_{n \mathop \to \infty} x_n$ the limit inferior of $\sequence {x_n}$. Then: :$\displaystyle \overline l = \limsup_{n \mathop \to \infty} x_n = \map {\lim_{n \mathop \to \infty} } {\sup_{k \mathop \ge n} x_k}$ :$\displaystyle \underline l = \liminf_{n \mathop \to \infty} x_n = \map {\lim_{n \mathop \to \infty} } {\inf_{k \mathop \ge n} x_k}$


Nth Root Test

Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a series of real numbers $\R$ or complex numbers $\C$. Let the sequence $\sequence {a_n}$ be such that the limit superior $\displaystyle \limsup_{n \mathop \to \infty} \size {a_n}^{1/n} = l$. Then: :If $l > 1$, the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ diverges. :If $l < 1$, the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ converges absolutely.


Series of Power over Factorial Converges

The series $\displaystyle \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}$ converges for all real values of $x$.


Area of Triangle in Terms of Side and Altitude

The area of a triangle $\triangle ABC$ is given by: :$\dfrac {c \cdot h_c} 2 = \dfrac {b \cdot h_b} 2 = \dfrac {a \cdot h_a} 2$ where: :$a, b, c$ are the sides :$h_a, h_b, h_c$ are the altitudes from $A$, $B$ and $C$ respectively.


Absolutely Convergent Series is Convergent

Let $V$ be a Banach space with norm $\norm {\, \cdot \,}$. Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be an absolutely convergent series in $V$. Then $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ is convergent.


Limit Comparison Test

Let $\left \langle {a_n} \right \rangle$ and $\left \langle {b_n} \right \rangle$ be sequences in $\R$. Let $\displaystyle \frac {a_n}{b_n} \to l$ as $n \to \infty$ where $l \in \R_{>0}$. Then the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ and $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ are either both convergent or both divergent.


Area of Parallelogram

The area of a parallelogram equals the product of one of its bases and the associated altitude.


Area of Triangle in Terms of Inradius and Exradii

The area of a $\triangle ABC$ is given by the formula: :$(ABC) = \rho_a \left({s - a}\right) = \rho_b \left({s - b}\right) = \rho_c \left({s - c}\right) = \rho s = \sqrt {\rho_a \rho_b \rho_c \rho}$ where: :$s$ is the semiperimeter :$I$ is the incenter :$\rho$ is the inradius :$I_a, I_b, I_c$ are the excenters :$\rho_a, \rho_b, \rho_c$ are the exradii from $I_a, I_b, I_c$, respectively.


Area of Triangle in Terms of Circumradius

Let $\triangle ABC$ be a triangle whose sides are of lengths $a, b, c$. Then the area $\AA$ of $\triangle ABC$ is given by: :$\AA = \dfrac {a b c} {4 R}$ where $R$ is the circumradius of $\triangle ABC$.


Stewart's Theorem

Let $\triangle ABC$ be a triangle with sides $a, b, c$. Let $CP$ be a cevian from $C$ to $P$. :400px Then: :$a^2 \cdot AP + b^2 \cdot PB = c \paren {CP^2 + AP \cdot PB}$


Length of Median of Triangle

Let $\triangle ABC$ be a triangle. Let $CD$ be the median of $\triangle ABC$ which bisects $AB$. :400px The length $m_c$ of $CD$ is given by: :${m_c}^2 = \dfrac {a^2 + b^2} 2 - \dfrac {c^2} 4$


Length of Angle Bisector

Let $\triangle ABC$ be a triangle. Let $AD$ be the angle bisector of $\angle BAC$ in $\triangle ABC$. :300px Let $d$ be the length of $AD$. Then $d$ is given by: :$d^2 = \dfrac {b c} {\paren {b + c}^2} \paren {\paren {b + c}^2 - a^2}$ where $a$, $b$, and $c$ are the sides opposite $A$, $B$ and $C$ respectively.


Supremum of Subset

Let $\left({U, \preceq}\right)$ be an ordered set. Let $S \subseteq U$. Let $T \subseteq S$. Let $S$ admit a supremum (in $U$). If $T$ also admits a supremum (in $U$), then $\sup \left({T}\right) \preceq\sup \left({S}\right)$.


Area of Triangle

This page gathers a variety of formulas for the area of a triangle.


Intersecting Chord Theorem

Let $AC$ and $BD$ both be chords of the same circle. Let $AC$ and $BD$ intersect at $E$. Then $AE \cdot EC = DE \cdot EB$. {{EuclidSaid}} :''If in a circle two straight lines cut one another, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other.'' {{EuclidPropRef|III|35}}


Supremum Plus Constant

Let $S$ be a subset of the set of real numbers $\R$. Let $S$ be bounded above. Let $\xi \in \R$. Then: :$\displaystyle \map {\sup_{x \mathop \in S} } {x + \xi} = \xi + \map {\sup_{x \mathop \in S} } x$ where $\sup$ denotes supremum.


Suprema and Infima of Combined Bounded Functions

Let $f$ and $g$ be real functions. Let $c$ be a constant.


Multiple of Infimum

Let $T \subseteq \R: T \ne \varnothing$ be a non-empty subset of the set of real numbers $\R$. Let $T$ be bounded below. Let $z \in \R: z > 0$ be a (strictly) positive real number. Then: :$\displaystyle \map {\inf_{x \mathop \in T} } {z x} = z \, \map {\inf_{x \mathop \in T} } x$ where $\inf$ denotes infimum.


Law of Sines

For any triangle $\triangle ABC$: :$\dfrac a {\sin A} = \dfrac b {\sin B} = \dfrac c {\sin C} = 2 R$ where: :$a$, $b$, and $c$ are the sides opposite $A$, $B$ and $C$ respectively :$R$ is the circumradius of $\triangle ABC$.


Negative of Infimum is Supremum of Negatives

Let $T$ be a non-empty subset of the real numbers $\R$. Let $T$ be bounded below. Then: : $(1): \quad \set {x \in \R: -x \in T}$ is bounded above : $(2): \quad \displaystyle -\inf_{x \mathop \in T} x = \map {\sup_{x \mathop \in T} } {-x}$ where $\sup$ and $\inf$ denote the supremum and infimum respectively.


Infimum Plus Constant

Let $T$ be a subset of the set of real numbers. Let $T$ be bounded below. Let $\xi \in \R$. Then: :$\displaystyle \map {\inf_{x \mathop \in T} } {x + \xi} = \xi + \map {\inf_{x \mathop \in T} } x$ where $\inf$ denotes infimum.


Infimum of Subset

Let $\struct {U, \preceq}$ be an ordered set. Let $S \subseteq U$. Let $T \subseteq S$. Let $\struct {S, \preceq}$ admit an infimum in $U$. If $T$ also admits an infimum in $U$, then $\map \inf S \preceq \map \inf T$.


Construction of Equilateral Triangle

On a given straight line segment, it is possible to construct an equilateral triangle. {{:Euclid:Proposition/I/1}}


Construction of Equal Straight Line

At a given point, it is possible to construct a straight line segment of length equal to that of any given straight line segment. The given point will be an endpoint of the constructed straight line segment. {{:Euclid:Proposition/I/2}}


Construction of Equal Straight Lines from Unequal

Given two unequal straight line segments, it is possible to cut off from the greater a straight line segment equal to the lesser. {{:Euclid:Proposition/I/3}}


Triangle Side-Angle-Side Equality

If $2$ triangles have: : $2$ sides equal to $2$ sides respectively : the angles contained by the equal straight lines equal they will also have: : their third sides equal : the remaining two angles equal to their respective remaining angles, namely, those which the equal sides subtend.


Isosceles Triangle has Two Equal Angles

In isosceles triangles, the angles at the base are equal to each other. Also, if the equal straight lines are extended, the angles under the base will also be equal to each other. {{:Euclid:Proposition/I/5}}


Triangle with Two Equal Angles is Isosceles

If a triangle has two angles equal to each other, the sides which subtend the equal angles will also be equal to one another. Hence, by definition, such a triangle will be isosceles. {{:Euclid:Proposition/I/6}}


Two Lines Meet at Unique Point

Let two straight line segments be constructed on a straight line segment from its endpoints so that they meet at a point. Then there cannot be two other straight line segments equal to the former two respectively, constructed on the same straight line segment and on the same side of it, meeting at a different point. {{:Euclid:Proposition/I/7}}


Triangle Side-Side-Side Equality

Let two triangles have all $3$ sides equal. Then they also have all $3$ angles equal. Thus two triangles whose sides are all equal are themselves congruent.


Bisection of Angle

It is possible to bisect any given rectilineal angle. {{:Euclid:Proposition/I/9}}


Bisection of Straight Line

It is possible to bisect a straight line segment. {{:Euclid:Proposition/I/10}}


External Angle of Triangle Greater than Internal Opposite

The external angle of a triangle is greater than either of the opposite internal angles. {{:Euclid:Proposition/I/16}}


Limit iff Limits from Left and Right

Let $f$ be a real function defined on an open interval $\openint a b$ except possibly at a point $c \in \openint a b$. Then: :$\map f x \to l$ as $x \to c$ {{iff}}: :$\map f x \to l$ as $x \to c^-$ and :$\map f x \to l$ as $x \to c^+$


Limit of Function by Convergent Sequences

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $S \subseteq A_1$ be an open set of $M_1$. Let $f$ be a mapping defined on $S$, except possibly at the point $c \in S$. Then $\displaystyle \lim_{x \mathop \to c} f \left({x}\right) = l$ iff: : for each sequence $\left \langle {x_n} \right \rangle$ of points of $S$ such that $\forall n \in \N_{>0}: x_n \ne c$ and $\displaystyle \lim_{n \to \infty} x_n = c$ it is true that: : $\displaystyle \lim_{n \to \infty} f \left({x_n}\right) = l$


Combination Theorem for Limits of Functions

Let $X$ be one of the standard number fields $\Q, \R, \C$. Let $f$ and $g$  be functions defined on an open subset $S \subseteq X$, except possibly at the point $c \in S$. Let $f$ and $g$ tend to the following limits: :$\displaystyle \lim_{x \to c} \ f \left({x}\right) = l$ :$\displaystyle \lim_{x \to c} \ g \left({x}\right) = m$ Let $\lambda, \mu \in X$ be arbitrary numbers in $X$. Then the following results hold:


Real Polynomial Function is Continuous

A (real) polynomial function is continuous at every point. Thus a (real) polynomial function is continuous on every interval of $\R$.


Linear Function is Continuous

Let $\alpha, \beta \in \R$ be real numbers. Let $f : \R \to \R$ be the real function with: :$\map f x = \alpha x + \beta$ for all $x \in \R$. Then $f$ is continuous at every real number $c \in \R$.


Real Rational Function is Continuous

A real rational function is continuous at every point at which it is defined. Thus a real rational function is continuous on every interval of $\R$ not containing a root of the denominator of the function.


Limit of Composite Function

Let $f$ and $g$ be real functions. Let: :$\displaystyle \lim_{y \mathop \to \eta} \map f y = l$ :$\displaystyle \lim_{x \mathop \to \xi} \map g x = \eta$ Then, if either: :'''Hypothesis 1:''' $f$ is continuous at $\eta$ (that is $l = \map f \eta$) or: :'''Hypothesis 2:''' for some open interval $I$ containing $\xi$, it is true that $\map g x \ne \eta$ for any $x \in I$ except possibly $x = \xi$ then: : $\displaystyle \lim_{x \mathop \to \xi} \map f {\map g x} = l$


Limit of Image of Sequence

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces. Let $f: A_1 \to A_2$ be a mapping which is continuous at $a \in A_1$. Let $\sequence {x_n}$ be a sequence of points in $A_1$ such that: :$\displaystyle \lim_{n \mathop \to \infty} x_n = a$ where $\displaystyle \lim_{n \mathop \to \infty} x_n$ is the limit of $x_n$. Then: :$\displaystyle \lim_{n \mathop \to \infty} \map f {x_n} = \map f a$ That is: :$\displaystyle \lim_{n \mathop \to \infty} \map f {x_n} = \map f {\lim_{n \mathop \to \infty} x_n}$ That is, for a continuous mapping, the limit and function symbols commute.


Image of Interval by Continuous Function is Interval

Let $I$ be a real interval. Let $f: I \to \R$ be a continuous real function. Then the image of $f$ is a real interval.


Interval Divided into Subsets

Let $\mathbb I$ be a real interval. Let $S$ and $T$ be non-empty subsets of $\mathbb I$ such that $\mathbb I \subseteq S \cup T$. Then one of $S$ or $T$ contains an element at zero distance from the other.