hcju/proofwikips huggingface dataset:
Definition:Kilogram
The '''kilogram''' is the SI base unit of mass. It is defined as being equal to: :The fixed numerical value of the Planck constant $h$ to be $6 \cdotp 62607015 \times 10^{-34}$ when expressed in the unit Joule seconds. The Joule second is equal to $1 \ \mathrm {kg} \mathrm m^2 \mathrm s^{−1}$, where the metre and the second are defined in terms of: :the speed of light $c$ :the time of transition between the two hyperfine levels of the ground state of the caesium $133$ atom at rest at $0 \ \mathrm K$.
Definition:Gram
The '''gram''' is the CGS base unit of mass. It is defined as the mass of a cubic centimetre water at the temperature of melting ice.
Definition:Temperature
'''Temperature''' is a physical property of matter that quantifies how hot or cold a body is. It is a scalar quantity which can be mapped directly to the real number line.
Definition:Kelvin
The '''kelvin''' is the SI base unit of temperature. Its two reference points are: :$0 \ \mathrm K$, which is set at absolute zero, the temperature at which all thermal motion stops :$273 \cdotp 16 \ \mathrm K$, which is set at the {{WP|Triple_point|triple point}} of water.
Definition:Absolute Zero
'''Absolute zero''' is the lowest temperature which can theoretically be achieved. It is the temperature where all motion due to thermal effects stops. Before that temperature can be reached, quantum effects come into play.
Definition:Electric Charge
'''Electric charge''' is a physical quantity of matter which causes it to experience a force when near other electrically charged matter. It is a scalar quantity.
Definition:Electric Current
'''Electric current''' is the physical process caused by the flow of electrically charged particles, usually electrons. Thus it is defined as the rate of flow of electric charge: :$I = \dfrac {\d Q} {\d t}$
Definition:Ampere (Unit)
The '''ampere''' is the SI base unit of electric current. It is defined as being: :The constant current which will produce a force of attraction whose value is $2 \times 10^{–7}$ newtons per metre of length between two straight, parallel conductors of infinite length and negligible circular cross section placed one metre apart in a vacuum. This arises from Ampère's Force Law, which states that such an attractive force exists. === Symbol === {{:Symbols:A/Ampere}} {{NamedforDef|André-Marie Ampère|cat = Ampère}}
Definition:Newton (Unit)
The '''newton''' is the SI unit of force. It is defined as being: :The amount of force required to accelerate a mass of one kilogram at a rate of one metre per second squared.
Definition:Coulomb (Unit)
The '''coulomb''' is the SI unit of electric charge. It is defined as being: :The amount of electric charge carried in one second by a constant current of one ampere.
Definition:Cube/Geometry
A '''cube''' is a hexahedron whose $6$ faces are all congruent squares.
Definition:CGS
'''CGS''' is the centimetre-gram-second standard system of units of measurement. This system is rarely used nowadays, the SI units having largely taken over.
Definition:MKS
'''MKS''' is the metre-kilogram-second standard system of units of measurement.
Definition:FPS
'''FPS''' is the foot-pound-second standard system of units of measurement. It is derived from the imperial system, and was used throughout the British Empire until the mid-20th century.
Definition:Imperial
The '''imperial system''' is a system of measurement based on traditional established folk measures, adopted by the British empire and used in Britain until the mid-20th century. Its base units can be understood as being the FPS Base Units
Definition:Free Fall
A body $B$ influenced by a gravitational field $M$ is in '''free fall'''  the force on it caused by $M$ is the only force on $B$.
Definition:Gravitational Field
Every body which has mass influences every other body which has mass, according to Newton's Law of Universal Gravitation. Thus any body can be considered as being surrounded by a field given rise to by its mass, called a '''gravitational field'''. Its value $\mathbf g$ at any point is given by: :$\mathbf g = \dfrac {G M} {d^3} \mathbf d$ where: :$G$ is the gravitational constant; :$M$ is the mass of the body; :$\mathbf d$ is the displacement vector from the point to the center of gravity of the body, whose magnitude is $d$.
Definition:Field (Physics)
A '''field''', in the context of physics, is a physical quantity associated with every point of spacetime. The physical quantity in question may be either in vector or scalar form.
Definition:Spacetime
'''Spacetime''' (or''' space–time''', or '''space/time''') is a mathematical model that combines space and time into a single object. The form this model takes is often a four-dimensional vector space: three dimensions for space and one for time. More complicated models are used for more specialized applications.  ibycmme5pk6pp2aeslpiylu7v6owaax
Definition:Cube Root
{{:Definition:Cube Root/Real}}
Definition:Unit Vector
Let $\mathbf v$ be a vector quantity. The '''unit vector''' in the direction of $\mathbf v$ is defined and denoted as: :$\mathbf {\hat v} = \dfrac {\mathbf v} {\size {\mathbf v} }$ where $\size {\mathbf v}$ is the magnitude of $\mathbf v$.
Definition:Spring
A '''spring''' is a mechanical device for storing elastic potential energy. It works by resisting a force which acts so as to deform it. The greater the deformation, the greater the force. === Ideal Spring === {{:Definition:Spring/Ideal}} === Equilibrium Position === {{:Definition:Spring/Equilibrium Position}}
Definition:Density (Physics)
'''Density''' is a physical quantity. The '''density''' of a body is its mass per unit volume.
Definition:Volume
'''Volume''' is the measure of the extent of a body. It has three dimensions and is specified in units of length cubed.
Definition:Homogeneous (Physics)
A body is said to be '''homogeneous'''  the substance of any part of it is indistinguishable from any other part.
Definition:Weight (Physics)
The '''weight''' of a body is the magnitude of the force exerted on it by the influence of a gravitational field. The context is that the gravitational field in question is usually that of the Earth.
Definition:Chain (Physics)
A '''chain''' is an inelastic thread whose stiffness and width are approximated to zero. The mass of a '''chain''' is usually defined in terms of linear density.
Definition:Cord (Physics)
A '''cord''' is an inelastic thread whose mass, stiffness and width are approximated to zero. That is, it is a chain whose mass is zero.
Definition:Cone
A '''cone''' is a three-dimensional geometric figure which consists of the set of all straight lines joining the boundary of a plane figure $PQR$ to a point $A$ not in the same plane of $PQR$: :300px
Definition:Right Circular Cone
A '''right circular cone''' is a cone: : whose base is a circle : in which there is a line perpendicular to the base through its center which passes through the apex of the cone: : which is made by having a right-angled triangle turning along one of the sides that form the right angle. :300px  :''{{:Definition:Euclid's Definitions - Book XI/18 - Cone}}'' {{EuclidDefRefNocat|XI|18|Cone}}
Definition:Cross-Section
A '''cross-section''' of a 3-dimensional figure $F$ is the intersection of $F$ with a plane.  ogr1fkh0dxg0lfa72ibeihamehx0jxp
Definition:Power of Mapping
Let $f: S \to S$ be a mapping from $S$ to itself. Because the domain of $f$ is equal to the codomain of $f$ (both are $S$), the composite mapping $f \circ f$ is defined. We define the '''$n$th power of $f$''' as: :$\forall n \in \N: f^n = \begin{cases} I_S & : n = 0 \\ f \circ f^{n - 1} & : n > 0 \end{cases}$ where $I_S$ is the identity mapping.
Definition:Initial Segment of Natural Numbers
=== Zero-Based === {{:Definition:Initial Segment of Natural Numbers/Zero-Based}} === One-Based === {{:Definition:Initial Segment of Natural Numbers/One-Based}}
Definition:Count
To '''count''' a set $S$ is to establish a bijection between $S$ and an initial segment $\N_n$ of the natural numbers $\N$. If $S \sim \N_n$ (where $\sim$ denotes set equivalence) then we have '''counted''' $S$ and found it has $n$ elements. If $S \sim \N$ then $S$ is infinite but countable.
Definition:Multiplicative Notation
:$x y$ is used to indicate the result of the operation on $x$ and $y$. There is no symbol used to define the operation itself. :$e$ or $1$ is used for the identity element. :$x^{-1}$ is used for the inverse element. :$x^n$ is used to indicate the $n$th power of $x$.
Definition:Additive Notation
:$x + y$ is used to indicate the result of the operation $+$ on $x$ and $y$. :$e$ or $0$ is used for the identity element. Note that in this context, $0$ is '''not''' a zero element. :$-x$ is used for the inverse element. :$n x$ is used to indicate the $n$th power of $x$.
Definition:Set of Integer Multiples
The set $n \Z$ is defined as: :$\set {x \in \Z: n \divides x}$ for some $n \in \Z_{>0}$. That is, it is the set of all integers which are divisible by $n$, that is, the '''set of integer multiples of $n$'''. Thus we have: :$n \Z = \set {\ldots, -3 n, -2 n, -n, 0, n, 2 n, 3 n, \ldots}$
Definition:Transversal (Group Theory)
Let $G$ be a group. Let $H$ be a subgroup of $G$. Let $S \subseteq G$ be a subset of $G$. === Left Transversal === {{:Definition:Transversal (Group Theory)/Left Transversal}} === Right Transversal === {{:Definition:Transversal (Group Theory)/Right Transversal}} === Transversal === A '''transversal for $H$ in $G$''' is either a left transversal or a right transversal.
Definition:Product of Differences
Let $n \in \Z_{> 0}$ be a strictly positive integer. Let $\tuple {x_1, x_2, \ldots, x_n}$ be an ordered $n$-tuple of real numbers. The '''product of differences''' of $\tuple {x_1, x_2, \ldots, x_n}$ is defined and denoted as: :$\map {\Delta_n} {x_1, x_2, \ldots, x_n} = \displaystyle \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_i - x_j}$ When the underlying ordered $n$-tuple is understood, the notation is often abbreviated to $\Delta_n$. Thus $\Delta_n$ is the product of the difference of all ordered pairs of $\tuple {x_1, x_2, \ldots, x_n}$ where the index of the first is less than the index of the second.
Definition:Ordinary Space
'''Ordinary space''' (or just '''space''') is a word used to mean '''the universe we live in'''. The intuitive belief is that space is $3$-dimensional and therefore isomorphic to the real vector space $\R^3$. Hence '''ordinary space''' is usually taken as an alternative term for Euclidean $3$-dimensional space.
Definition:Computational Method
A '''computational method''' is an ordered quadruple $\left({Q, I, \Omega, f}\right)$ in which: : $Q$ is a set representing the '''states of the computation''' : $I$ is a set representing the '''input to the computation''' : $\Omega$ is a set representing the '''output from the computation''' : $f: Q \to Q$ is a mapping representing the '''computational rule''' subject to the following constraints: : $I \subseteq Q$ and $\Omega \subseteq Q$ : $\forall x \in \Omega: f \left({x}\right) = x$ === Computational Sequence === {{:Definition:Computational Method/Computational Sequence}}
Definition:Set of All Linear Transformations
Let $G$ and $H$ be $R$-modules. Then $\map {\mathrm {Hom}_R} {G, H}$ is the '''set of all linear transformations''' from $G$ to $H$: :$\map {\mathrm {Hom}_R} {G, H} := \set {\phi: G \to H: \phi \mbox{ is a linear transformation} }$ If it is clear (and therefore does not need to be stated) that the scalar ring is $R$, then this can be written $\map {\mathrm {Hom} } {G, H}$. Similarly, $\map {\mathrm {Hom}_R} G$ is the set of all linear operators on $G$: :$\map {\mathrm {Hom}_R} G := \set {\phi: G \to G: \phi \text{ is a linear operator} }$ Again, this can also be written $\map {\mathrm {Hom} } G$.
Definition:Inverse Matrix
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\mathbf A$ be a square matrix of order $n$. Let there exist a square matrix $\mathbf B$ of order $n$ such that: :$\mathbf A \mathbf B = \mathbf I_n = \mathbf B \mathbf A$ where $\mathbf I_n$ denotes the unit matrix of order $n$. Then $\mathbf B$ is called the '''inverse of $\mathbf A$''' and is usually denoted $\mathbf A^{-1}$.
Definition:Stirling's Triangle of the Second Kind
$\begin{array}{r|rrrrrrrrrr} n & {n \brace 0} & {n \brace 1} & {n \brace 2} & {n \brace 3} & {n \brace 4} & {n \brace 5} & {n \brace 6} & {n \brace 7} & {n \brace 8} & {n \brace 9} \\ \hline 0 & 1 & 0 &   0 &    0 &    0 &    0 &    0 &   0 &  0  & 0 \\ 1 & 0 & 1 &   0 &    0 &    0 &    0 &    0 &   0 &  0  & 0 \\ 2 & 0 & 1 &   1 &    0 &    0 &    0 &    0 &   0 &  0  & 0 \\ 3 & 0 & 1 &   3 &    1 &    0 &    0 &    0 &   0 &  0  & 0 \\ 4 & 0 & 1 &   7 &    6 &    1 &    0 &    0 &   0 &  0  & 0 \\ 5 & 0 & 1 &  15 &   25 &   10 &    1 &    0 &   0 &  0  & 0 \\ 6 & 0 & 1 &  31 &   90 &   65 &   15 &    1 &   0 &  0  & 0 \\ 7 & 0 & 1 &  63 &  301 &  350 &  140 &   21 &   1 &  0  & 0 \\ 8 & 0 & 1 & 127 &  966 & 1701 & 1050 &  266 &  28 &  1  & 0 \\ 9 & 0 & 1 & 255 & 3025 & 7770 & 6951 & 2646 & 462 & 36  & 1 \\ \end{array}$
Definition:Stirling's Triangle of the First Kind (Signed)
$\begin{array}{r|rrrrrrrrrr} n & \map s {n, 0} & \map s {n, 1} & \map s {n, 2} & \map s {n, 3} & \map s {n, 4} & \map s {n, 5} & \map s {n, 6} & \map s {n, 7} & \map s {n, 8} & \map s {n, 9} \\ \hline 0 & 1 &     0 &       0 &      0 &      0 &     0 &     0 &   0 &   0  & 0 \\ 1 & 0 &     1 &       0 &      0 &      0 &     0 &     0 &   0 &   0  & 0 \\ 2 & 0 &    -1 &       1 &      0 &      0 &     0 &     0 &   0 &   0  & 0 \\ 3 & 0 &     2 &      -3 &      1 &      0 &     0 &     0 &   0 &   0  & 0 \\ 4 & 0 &    -6 &      11 &     -6 &      1 &     0 &     0 &   0 &   0  & 0 \\ 5 & 0 &    24 &     -50 &     35 &    -10 &     1 &     0 &   0 &   0  & 0 \\ 6 & 0 &  -120 &     274 &   -225 &     85 &   -15 &     1 &   0 &   0  & 0 \\ 7 & 0 &   720 &   -1764 &   1624 &   -735 &   175 &   -21 &   1 &   0  & 0 \\ 8 & 0 & -5040 &   13068 & -13132 &   6769 & -1960 &   322 & -28 &   1  & 0 \\ 9 & 0 & 40320 & −109584 & 118124 & −67284 & 22449 & −4536 & 546 & −36  & 1 \\ \end{array}$
Definition:Root of Unity
Let $n \in \Z_{> 0}$ be a strictly positive integer. Let $F$ be a field. The '''$n$th roots of unity of $F$''' are defined as: :$U_n = \left\{{z \in F: z^n = 1}\right\}$
Definition:Internal Group Direct Product
Let $\struct {H_1, \circ {\restriction_{H_1} } }, \struct {H_2, \circ {\restriction_{H_2} } }$ be subgroups of a group $\struct {G, \circ}$ where $\circ {\restriction_{H_1} }, \circ {\restriction_{H_2} }$ are the restrictions of $\circ$ to $H_1, H_2$ respectively. === Definition 1 === {{:Definition:Internal Group Direct Product/Definition 1}} === Definition 2 === {{:Definition:Internal Group Direct Product/Definition 2}} === Definition 3 === {{:Definition:Internal Group Direct Product/Definition 3}}
Definition:Permutable Subgroups
Let $\struct {G, \circ}$ be a group. Let $H$ and $K$ be subgroups of $G$. Let $H \circ K$ denote the subset product of $H$ and $K$. Then $H$ and $K$ are '''permutable''' : :$H \circ K = K \circ H$
Definition:Cycle Type
Let $S_n$ denote the symmetric group on $n$ letters. Let $\rho \in S_n$. From Existence and Uniqueness of Cycle Decomposition, every $\rho$ may be uniquely expressed as a product of disjoint cycles: :$\rho = \tau_1, \tau_2, \ldots, \tau_r$ up to the order of factors. Let $\tau_1, \tau_2, \ldots, \tau_r$ be arranged in increasing order of cycle length. Let the length of the cycle $\tau_i$ be $k_i$. The resulting ordered tuple of cycle lengths $\tuple {k_1, k_2, \ldots, k_r}$ is called the '''cycle type''' of $\rho$. Thus $\sigma$ and $\rho$ have the same '''cycle type''' if they have the same number of cycles of equal length.
Definition:Permutation on Polynomial
Let $\map f {x_1, x_2, \ldots, x_n}$ denote a polynomial in $n$ variables $x_1, x_2, \ldots, x_n$. Let $S_n$ denote the symmetric group on $n$ letters. Let $\pi, \rho \in S_n$. Then $\pi * f$ is the polynomial obtained by applying the permutation $\pi$ to the subscripts on the variables of $f$. This is called the '''permutation on the polynomial $f$ by $\pi$''', or the $f$-permutation by $\pi$.
Definition:Pullback of Quotient Group Isomorphism
Let $G, H$ be groups. Let $N \lhd G, K \lhd H$ be normal subgroups of $G$ and $H$ respectively. Let: :$G / N \cong H / K$ where: :$G / N$ denotes the quotient of $G$ by $N$ :$\cong$ denotes group isomorphism. Let $\theta: G / N \to H / K$ be such a group isomorphism. The '''pullback $G \times^\theta H$ of $G$ and $H$ via $\theta$''' is the subset of $G \times H$ of elements of the form $\tuple {g, h}$ where $\map \theta {g N} = h K$.
Definition:Permutation Group
A '''permutation group''' on a set $S$ is a subgroup of the symmetric group $\struct {\map \Gamma S, \circ}$ on $S$.
Definition:Commutative and Unitary Ring
A '''commutative and unitary ring''' $\struct {R, +, \circ}$ is a ring with unity which is also commutative. That is, it is a ring such that the ring product $\struct {R, \circ}$ is commutative and has an identity element. That is, such that the multiplicative semigroup $\struct {R, \circ}$ is a commutative monoid.
Definition:K-Algebra
Let $k$ be a field. A '''$k$-algebra''' is a unital associative commutative algebra over $k$.
Definition:Entropic Operation
Let $\circ$ be an operation on a set $S$ such that: :$\forall a, b, c, d \in S: \left({a \circ b}\right) \circ \left({c \circ d}\right) = \left({a \circ c}\right) \circ \left({b \circ d}\right)$ Then $\circ$ is '''entropic'''.
Definition:Finite Ring Homomorphism
Let $A$ and $B$ be commutative rings with unity. Let $\phi: A \to B$ be a ring homomorphism. === Definition 1 === $\phi$ is '''finite'''  $B$ is finite as an algebra over $A$ via $\phi$. === Definition 2 === $\phi$ is '''finite'''  there exists a finite number of $b_1, \ldots, b_n$ such that every $b \in B$ can be written as: :$\displaystyle b = \sum_{i \mathop = 1}^n \phi \left({a_i}\right) b_i$ where $a_i \in A$.
Definition:Composition (Combinatorics)
A $k$-composition of a (strictly) positive integer $n \in \Z_{> 0}$ is an ordered $k$-tuple: :$c = \left({c_1, c_2, \ldots, c_k}\right)$ such that: : $(1): \quad c_1 + c_2 + \cdots + c_k = n$ : $(2): \quad \forall i \in \left[{1 \,.\,.\, k}\right]: c_i \in \Z_{> 0}$, that is, all the $c_i$ are strictly positive integers.
Definition:Ideal of Ring/Proper Ideal
A '''proper ideal''' $J$ of $\left({R, +, \circ}\right)$ is an ideal of $R$ such that $J$ is a proper subset of $R$. That is, such that $J \subseteq R$ and $J \ne R$.
Definition:Generated Ideal of Ring
Let $\struct {R, +, \circ}$ be a ring. Let $S \subseteq R$ be a subset. === Definition 1 === {{Definition:Generated Ideal of Ring/Definition 1}} === Definition 2: for commutative rings with unity === {{Definition:Generated Ideal of Ring/Definition 2}} === Definition 3: for rings with unity === {{Definition:Generated Ideal of Ring/Definition 3}}
Definition:Internal Direct Sum of Rings
Let $\left({R, +, \circ}\right)$ be a ring. Let $S_1, S_2, \ldots, S_n$ be a finite sequence of subrings of $R$. Let $\displaystyle S = \prod_{j \mathop = 1}^n S_j$ be the cartesian product of $S_1$ to $S_n$. Then $S$ is the '''(ring) direct sum''' of $S_1, S_2, \ldots, S_n$  the mapping $\phi: S \to R$ defined as: :$\phi\left({\left({x_1, x_2, \ldots, x_n}\right)}\right) = x_1 + x_2 + \cdots x_n$ is an isomorphism from $S$ to $R$.
Definition:Prime Ideal of Ring
Let $R$ be a ring. A '''prime ideal''' of $R$ is a proper ideal $P$ such that: : $I \circ J \subseteq P \implies I \subseteq P \text { or } J \subseteq P$ for any ideals $I$ and $J$ of $R$.
Definition:Generated Division Subring
Let $\left({D, +, \circ}\right)$ be a division ring. Let $S \subseteq D$. The '''division subring generated by $S$''' is the smallest division subring of $D$ containing $S$.
Definition:Isomorphism (Abstract Algebra)/Field Isomorphism
Let $\struct {F, +, \circ}$ and $\struct {K, \oplus, *}$ be fields. Let $\phi: F \to K$ be a (field) homomorphism. Then $\phi$ is a field isomorphism  $\phi$ is a bijection.
Definition:Cossist
The '''cossist''' tradition was a school of abacist mathematics developed in Germany in the 15th century. Notable cossists include {{AuthorRef|Johann Faulhaber}}.
Definition:Scalar Multiplication
=== $R$-Algebraic Structure === {{:Definition:Scalar Multiplication/R-Algebraic Structure}} === Module === {{:Definition:Scalar Multiplication/Module}} === Vector Space === {{:Definition:Scalar Multiplication/Vector Space}}
Definition:R-Algebraic Structure Automorphism
Let $\left({S, \ast_1, \ast_2, \ldots, \ast_n, \circ}\right)_R$ be an $R$-algebraic structure. Let $\phi: S \to S$ be an $R$-algebraic structure isomorphism from $S$ to itself. Then $\phi$ is an $R$-algebraic structure automorphism. This definition continues to apply when $S$ is a module, and also when it is a vector space.
Definition:Generator of Module
Let $R$ be a ring. Let $M$ be an $R$-module. Let $S \subseteq M$ be a subset. === Definition 1 === {{:Definition:Generator of Module/Definition 1}} === Definition 2 === {{:Definition:Generator of Module/Definition 2}} === Definition 3 === {{:Definition:Generator of Module/Definition 3}}
Definition:Kernel of Linear Transformation
Let $\phi: G \to H$ be a linear transformation where $G$ and $H$ are $R$-modules. Let $e_H$ be the identity of $H$. The '''kernel''' of $\phi$ is defined as: :$\map \ker \phi := \phi^{-1} \sqbrk {\set {e_H} }$. where $\phi^{-1} \sqbrk S$ denotes the preimage of $S$ under $\phi$. === In Vector Space === {{:Definition:Kernel of Linear Transformation/Vector Space}}
Definition:R-Algebraic Structure Monomorphism
Let $\left({S, \ast_1, \ast_2, \ldots, \ast_n, \circ}\right)_R$ and $\left({T, \odot_1, \odot_2, \ldots, \odot_n, \otimes}\right)_R$ be $R$-algebraic structures. Then $\phi: S \to T$ is an '''$R$-algebraic structure monomorphism''' : : $(1): \quad \phi$ is an injection : $(2): \quad \forall k: k \in \left[{1 \,.\,.\, n}\right]: \forall x, y \in S: \phi \left({x \ast_k y}\right) = \phi \left({x}\right) \odot_k \phi \left({y}\right)$ : $(3): \quad \forall x \in S: \forall \lambda \in R: \phi \left({\lambda \circ x}\right) = \lambda \otimes \phi \left({x}\right)$. That is, : : $(1): \quad \phi$ is an injection : $(2): \quad \phi$ is an $R$-algebraic structure homomorphism. This definition continues to apply when $S$ and $T$ are modules, and also when they are vector spaces. === Vector Space Monomorphism === {{:Definition:Vector Space Monomorphism}}
Definition:Vector Space Monomorphism
Let $V$ and $W$ be $K$-vector spaces. Then $\phi: V \to W$ is a '''vector space monomorphism''' : : $(1): \quad \phi$ is an injection : $(2): \quad \forall \mathbf x, \mathbf y \in V: \phi \left({\mathbf x + \mathbf y}\right) = \phi \left({\mathbf x}\right) + \phi \left({\mathbf y}\right)$ : $(3): \quad \forall \mathbf x \in V: \forall \lambda \in K: \phi \left({\lambda \mathbf x}\right) = \lambda \phi \left({\mathbf x}\right)$
Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism/Vector Space Isomorphism
Let $\struct {V, +, \circ }$ and $\struct {W, +', \circ'}$ be $K$-vector spaces. Then $\phi: V \to W$ is a '''vector space isomorphism''' : : $(1): \quad \phi$ is a bijection : $(2): \quad \forall \mathbf x, \mathbf y \in V: \map \phi {\mathbf x + \mathbf y} = \map \phi {\mathbf x} +' \map \phi {\mathbf y}$ : $(3): \quad \forall \mathbf x \in V: \forall \lambda \in K: \map \phi {\lambda \mathbf x} = \lambda \map \phi {\mathbf x}$
Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism/Module Isomorphism
Let $\struct {G, +_G, \circ}_R$ and $\struct {H, +_H, \circ}_R$ be $R$-modules. Let $\phi: G \to H$ be a module homomorphism. Then $\phi$ is a '''module isomorphism'''  $\phi$ is a bijection.