hcju/proofwikips huggingface dataset:
Definition:Strictly Positive/Rational Number
The '''strictly positive rational numbers''' are the set defined as: :$\Q_{>0} := \set {x \in \Q: x > 0}$ That is, all the rational numbers that are strictly greater than zero.
Definition:Strictly Positive/Real Number
=== Definition 1 === {{:Definition:Strictly Positive/Real Number/Definition 1}} === Definition 2 === {{:Definition:Strictly Positive/Real Number/Definition 2}}
Definition:Negative/Number
The concept of negative can be applied to the following sets of numbers: :$(1): \quad$ The integers $\Z$ :$(2): \quad$ The rational numbers $\Q$ :$(3): \quad$ The real numbers $\R$ The Complex Numbers cannot be Ordered Compatibly with Ring Structure, so there is no such concept as a negative complex number. === Integers === {{:Definition:Negative/Integer}} === Rational Numbers === {{:Definition:Negative/Rational Number}} === Real Numbers === {{:Definition:Negative/Real Number}} === Complex Numbers === {{:Definition:Negative/Complex Number}}
Definition:Negative/Rational Number
The '''negative rational numbers''' are the set defined as: :$\Q_{\le 0} := \set {x \in \Q: x \le 0}$ That is, all the rational numbers that are less than or equal to zero.
Definition:Negative/Real Number
The '''negative real numbers''' are the set defined as: :$\R_{\le 0} := \set {x \in \R: x \le 0}$ That is, all the real numbers that are less than or equal to zero.
Definition:Negative/Integer
The '''negative integers''' comprise the set: :$\set {0, -1, -2, -3, \ldots}$ As the set of integers is the Inverse Completion of Natural Numbers, it follows that elements of $\Z$ are the isomorphic images of the elements of equivalence classes of $\N \times \N$ where two tuples are equivalent if the difference between the two elements of each tuple is the same. Thus '''negative''' can be formally defined on $\Z$ as a relation induced on those equivalence classes as specified in the definition of integers. That is, the integers being defined as all the difference congruence classes, '''negative''' can be defined directly as the relation specified as follows: The integer $z \in \Z: z = \eqclass {\tuple {a, b} } \boxminus$ is '''negative'''  $b > a$. The set of '''negative integers''' is denoted $\Z_{\le 0}$. An element of $\Z$ can be specifically indicated as being '''negative''' by prepending a $-$ sign: :$-x \in \Z_{\le 0} \iff x \in \Z_{\ge 0}$
Definition:Strictly Negative/Number
The concept of strictly negative can be applied to the following sets of numbers: :$(1): \quad$ The integers $\Z$ :$(2): \quad$ The rational numbers $\Q$ :$(3): \quad$ The real numbers $\R$ === Integers === {{:Definition:Strictly Negative/Integer}} === Rational Numbers === {{:Definition:Strictly Negative/Rational Number}} === Real Numbers === {{:Definition:Strictly Negative/Real Number}}
Definition:Strictly Negative/Integer
The '''strictly negative integers''' are the set defined as: {{begin-eqn}} {{eqn | l = \Z_{< 0}       | o = :=       | r = \set {x \in \Z: x < 0} }} {{eqn | r = \set {-1, -2, -3, \ldots} }} {{end-eqn}} That is, all the integers that are strictly less than zero.
Definition:Strictly Negative/Rational Number
The '''strictly negative rational numbers''' are the set defined as: :$\Q_{< 0} := \set {x \in \Q: x < 0}$ That is, all the rational numbers that are strictly less than zero.
Definition:Strictly Negative/Real Number
The '''strictly negative real numbers''' are the set defined as: :$\R_{<0} := \set {x \in \R: x < 0}$ That is, all the real numbers that are strictly less than zero.
Definition:Ordering Induced by Positivity Property
Let $\struct {R, +, \circ, \le}$ be an ordered ring whose zero is $0_R$ and whose unity is $1_R$. Let $P \subseteq R$ such that: :$(1): \quad P + P \subseteq P$ :$(2): \quad P \cap \paren {-P} = \set {0_R}$ :$(3): \quad P \circ P \subseteq P$ Then the ordering $\le$ compatible with the ring structure of $R$ is called the '''ordering induced by (the positivity property) $P$'''.
Definition:Rational Number/Formal Definition
The field $\struct {\Q, +, \times}$ of rational numbers is the field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers. This is shown to exist in Existence of Field of Quotients. In view of Field of Quotients is Unique, we construct the field of quotients of $\Z$, give it a label $\Q$ and call its elements '''rational numbers'''.
Definition:Ordered Ring Isomorphism
Let $\struct {S, +, \circ, \preceq}$ and $\struct {T, \oplus, *, \preccurlyeq}$ be ordered rings. An '''ordered ring isomorphism''' from $\struct {S, +, \circ, \preceq}$ to $\struct {T, \oplus, *, \preccurlyeq}$ is a mapping $\phi: S \to T$ that is both: :$(1): \quad$ An ordered group isomorphism from the ordered group $\struct {S, +, \preceq}$ to the ordered group $\struct {T, \oplus, \preccurlyeq}$ :$(2): \quad$ A semigroup isomorphism from the semigroup $\struct {S, \circ}$ to the semigroup $\struct {T, *}$.
Definition:Ordered Group Isomorphism
Let $\left({S, \circ, \preceq}\right)$ and $\left({T, *, \preccurlyeq}\right)$ be ordered groups. An '''ordered group isomorphism''' from $\left({S, \circ, \preceq}\right)$ to $\left({T, *, \preccurlyeq}\right)$ is a mapping $\phi: S \to T$ that is both: :$(1): \quad$ A group isomorphism from the group $\left({S, \circ}\right)$ to the group $\left({T, *}\right)$ :$(2): \quad$ An order isomorphism from the ordered set $\left({S, \preceq}\right)$ to the ordered set $\left({T, \preccurlyeq}\right)$.
Definition:Ordered Group Monomorphism
Let $\left({S, \circ, \preceq}\right)$ and $\left({T, *, \preccurlyeq}\right)$ be ordered groups. An '''ordered group monomorphism''' from $\left({S, \circ, \preceq}\right)$ to $\left({T, *, \preccurlyeq}\right)$ is a mapping $\phi: S \to T$ that is both: :$(1): \quad$ A group monomorphism from the group $\left({S, \circ}\right)$ to the group $\left({T, *}\right)$ :$(2): \quad$ An order embedding from the ordered set $\left({S, \preceq}\right)$ to the ordered set $\left({T, \preccurlyeq}\right)$.
Definition:Ordered Ring Monomorphism
Let $\left({S, +, \circ, \preceq}\right)$ and $\left({T, \oplus, *, \preccurlyeq}\right)$ be ordered rings. An '''ordered ring monomorphism''' from $\left({S, +, \circ, \preceq}\right)$ to $\left({T, \oplus, *, \preccurlyeq}\right)$ is a mapping $\phi: S \to T$ that is both: :$(1): \quad$ An ordered group monomorphism from the ordered group $\left({S,+, \preceq}\right)$ to the ordered group $\left({T, \oplus, \preccurlyeq}\right)$ :$(2): \quad$ A semigroup monomorphism from the semigroup $\left({S, \circ}\right)$ to the semigroup $\left({T, *}\right)$.
Definition:Ordered Structure Automorphism
Let $\left({S, \circ, \preceq}\right)$ be an ordered structures. Let $\phi: S \to S$ be an ordered structure isomorphism from $S$ to itself. Then $\phi$ is an ordered structure automorphism.
Definition:Ordered Group Automorphism
Let $\left({G, \circ, \preceq}\right)$ be an ordered group. An '''ordered group automorphism''' from $\left({G, \circ, \preceq}\right)$ to itself is a mapping $\phi: G \to G$ that is both: :$(1): \quad$ A group automorphism, that is, a group isomorphism from the group $\left({G, \circ}\right)$ to itself :$(2): \quad$ An order isomorphism from the ordered set $\left({G, \preceq}\right)$ to itself.
Definition:Ordered Ring Automorphism
Let $\struct {R, +, \circ, \preceq}$ be an ordered ring. An '''ordered ring automorphism''' from $\struct {R, +, \circ, \preceq}$ to itself is a mapping $\phi: R \to R$ that is both: :$(1): \quad$ An ordered group automorphism from the ordered group $\struct {R, +, \preceq}$ to itself :$(2): \quad$ A semigroup automorphism from the semigroup $\struct {R, \circ}$ to itself.
Definition:Abelian Group Axioms
{{begin-axiom}} {{axiom | n = \text G 0         | lc= Closure         | q = \forall x, y \in G         | m = x + y \in G }} {{axiom | n = \text G 1         | lc= Associativity         | q = \forall x, y, z \in G         | m = x + \paren {y + z} = \paren {x + y} + z }} {{axiom | n = \text G 2         | lc= Identity         | q = \exists 0 \in G: \forall x \in G         | m = 0 + x = x = x + 0 }} {{axiom | n = \text G 3         | lc= Inverse         | q = \forall x \in G: \exists \paren {-x}\in G         | m = x + \paren {-x} = 0 = \paren {-x} + x }} {{axiom | n = \text C         | lc= Commutativity         | q = \forall x, y \in G         | m = x + y = y + x }} {{end-axiom}}
Definition:Scalar/R-Algebraic Structure
The elements of the scalar ring $\struct {R, +_R, \times_R}$ are called '''scalars'''.
Definition:Scalar Multiplication/R-Algebraic Structure
Let $\struct {S, *_1, *_2, \ldots, *_n, \circ}_R$ be an $R$-algebraic structure with $n$ operations, where: :$\struct {R, +_R, \times_R}$ is a ring :$\struct {S, *_1, *_2, \ldots, *_n}$ is an algebraic structure with $n$ operations The operation $\circ: R \times S \to S$ is called '''scalar multiplication'''.
Definition:Scalar Ring/Zero Scalar
The zero of the scalar ring is called the '''zero scalar''' and usually denoted $0$, or, if it is necessary to distinguish it from the identity of $\struct {G, +_G}$, by $0_R$.
Definition:Vector Subspace/Proper Subspace
If $T$ is a proper subset of $S$, then $\struct {T, +_T, \circ_T}_K$ is a '''proper (vector) subspace''' of $\struct {S, +, \circ}_K$.
Definition:Vector Subspace/Hilbert Spaces
When considering Hilbert spaces, one wants to deal with projections onto subspaces. These projections however require the linear subspace to be closed in topological sense in order to be well-defined. Therefore, in treatises of Hilbert spaces, one encounters the terminology '''linear manifold''' for the concept of '''vector subspace''' defined above. The adapted definition of '''linear subspace''' is then that it is a topologically closed '''linear manifold'''.
Definition:Polynomial over Ring as Function on Free Monoid on Set
Let $M$ be the free commutative monoid on the indexed set $\left\{{X_j: j \in J}\right\}$. A '''polynomial''' in $\left\{{X_j: j \in J}\right\}$ can be defined as a mapping $f: M \to R$ of finite support. That is, it is an element of the ring of polynomial forms.
Definition:Polynomial Equation
A '''polynomial equation''' is an equation in the form: :$\map f x = 0$ where $f$ is a polynomial function.
Definition:Strongly Additive Function
Let $\mathcal S$ be an algebra of sets. Let $f: \mathcal S \to \overline{\R}$ be a function, where $\overline{\R}$ denotes the extended set of real numbers. Then $f$ is defined to be '''strongly additive''' : :$\forall S, T \in \mathcal S: f \left({S \cup T}\right) + f \left({S \cap T}\right) = f \left({S}\right) + f \left({T}\right)$
Definition:Polynomial Function/General Definition
Let $f = a_1 \mathbf X^{k_1} + \cdots + a_r \mathbf X^{k_r}$ be a polynomial form over $R$ in the indeterminates $\left\{{X_j: j \in J}\right\}$. For each $x = \left({x_j}\right)_{j \in J} \in R^J$, let $\phi_x: R \left[{\left\{{X_j: j \in J}\right\}}\right] \to R$ be the evaluation homomorphism from the ring of polynomial forms at $x$. Then the set: :$\left\{{\left({x, \phi_x \left({f}\right)}\right): x \in R^J}\right\} \subseteq R^J \times R$ defines a '''polynomial function''' $R^J \to R$.
Definition:Linear Combination/Sequence
Let $G$ be an $R$-module. Let $\left \langle {a_n} \right \rangle := \left \langle {a_j} \right \rangle_{1 \mathop \le j \mathop \le n}$ be a sequence of elements of $G$ of length $n$. An element $b \in G$ is a '''linear combination''' of $\left \langle {a_n} \right \rangle$ : : $\displaystyle \exists \left \langle {\lambda_n} \right \rangle \subseteq R: b = \sum_{k \mathop = 1}^n \lambda_k a_k$
Definition:Linear Combination/Subset
Let $G$ be an $R$-module. Let $\O \subset S \subseteq G$. Let $b \in G$ be a linear combination of some sequence $\sequence {a_n}$ of elements of $S$. Then $b$ is a '''linear combination of $S$'''.
Definition:Linear Combination/Empty Set
Let $G$ be an $R$-module. $b$ is a linear combination of $\varnothing$ : : $b = e_G$
Definition:Linearly Independent/Sequence
Let $\left \langle {a_n} \right \rangle$ be a sequence of elements of $G$ such that: : $\displaystyle \forall \left \langle {\lambda_n} \right \rangle \subseteq R: \sum_{k \mathop = 1}^n \lambda_k \circ a_k = e \implies \lambda_1 = \lambda_2 = \cdots = \lambda_n = 0_R$ That is, the only way to make $e$ with a linear combination of $\left \langle {a_n} \right \rangle$ is by making all the elements of $\left \langle {\lambda_n} \right \rangle$ equal to $0_R$. Such a sequence is '''linearly independent'''. === Linearly Independent Sequence on a Real Vector Space === {{:Definition:Linearly Independent/Sequence/Real Vector Space}}
Definition:Linearly Independent/Set
Let $S \subseteq G$. Then $S$ is a '''linearly independent set (over $R$)'''  every finite sequence of distinct terms in $S$ is a linearly independent sequence. That is, such that: : $\displaystyle \forall \sequence {\lambda_n} \subseteq R: \sum_{k \mathop = 1}^n \lambda_k \circ a_k = e \implies \lambda_1 = \lambda_2 = \cdots = \lambda_n = 0_R$ where $a_1, a_2, \ldots, a_k$ are distinct elements of $S$. === Linearly Independent Set on a Real Vector Space === {{:Definition:Linearly Independent/Set/Real Vector Space}} === Linearly Independent Set on a Complex Vector Space === {{:Definition:Linearly Independent/Set/Complex Vector Space}}
Definition:Linearly Independent/Sequence/Real Vector Space
Let $\struct {\R^n, +, \cdot}_\R$ be a real vector space. Let $\sequence {\mathbf v_n}$ be a sequence of vectors in $\R^n$. Then $\sequence {\mathbf v_n}$ is '''linearly independent''' : :$\ds \forall \sequence {\lambda_n} \subseteq \R: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0 \implies \lambda_1 = \lambda_2 = \cdots = \lambda_n = 0$ where $\mathbf 0 \in \R^n$ is the zero vector and $0 \in \R$ is the zero scalar.
Definition:Linearly Independent/Set/Real Vector Space
Let $\struct {\R^n, +, \cdot}_\R$ be a real vector space. Let $S \subseteq \R^n$. Then $S$ is a '''linearly independent set of real vectors''' if every finite sequence of distinct terms in $S$ is a linearly independent sequence. That is, such that: :$\displaystyle \forall \set {\lambda_k: 1 \le k \le n} \subseteq \R: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0 \implies \lambda_1 = \lambda_2 = \cdots = \lambda_n = 0$ where $\mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n$ are distinct elements of $S$.
Definition:Linearly Dependent/Sequence
Let $\left \langle {a_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ be a sequence of elements of $G$ such that: : $\displaystyle \exists \left \langle {\lambda_k} \right \rangle_{1 \mathop \le k \mathop \le n} \subseteq R: \sum_{k \mathop = 1}^n \lambda_k \circ a_k = e$ where not all of $\lambda_k$ are equal to $0_R$. That is, it is possible to find a linear combination of $\left \langle {a_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ which equals $e$. Such a sequence is '''linearly dependent'''. === Linearly Dependent Sequence on a Real Vector Space === {{:Definition:Linearly Dependent/Sequence/Real Vector Space}}
Definition:Linearly Dependent/Sequence/Real Vector Space
Let $\left({\R^n,+,\cdot}\right)_{\R}$ be a real vector space. Let $\mathbf 0 \in \R^n$ be the zero vector. Let $\left \langle {\mathbf v_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ be a sequence of vectors in $\R^n$. Then $\left \langle {\mathbf v_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ is '''linearly dependent''' iff: : $\displaystyle \exists \left \langle {\lambda_k} \right \rangle_{1 \mathop \le k \mathop \le n} \subseteq \R: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0$ where not all $\lambda_k$ are equal to $0$. That is, it is possible to find a linear combination of $\left \langle {\mathbf v_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ which equals $\mathbf 0$.
Definition:Linearly Dependent/Set
Let $S \subseteq G$. Then $S$ is a '''linearly dependent set''' if there exists a sequence of distinct terms in $S$ which is a linearly dependent sequence. That is, such that: :$\displaystyle \exists \set {\lambda_k: 1 \le k \le n} \subseteq R: \sum_{k \mathop = 1}^n \lambda_k \circ a_k = e$ where $a_1, a_2, \ldots, a_n$ are distinct elements of $S$, and where at least one of $\lambda_k$ is not equal to $0_R$.
Definition:Linearly Dependent/Set/Real Vector Space
Let $\struct {\R^n, +, \cdot}_\R$ be a real vector space. Let $S \subseteq \R^n$. Then $S$ is a '''linearly dependent set''' if there exists a sequence of distinct terms in $S$ which is a linearly dependent sequence. That is, such that: :$\displaystyle \exists \set {\lambda_k: 1 \le k \le n} \subseteq \R: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0$ where $\set {\mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n} \subseteq S$, and such that at least one of $\lambda_k$ is not equal to $0$.
Definition:Atom of Measure
Let $\struct {X, \Sigma, \mu}$ be a measure space. An element $x \in X$ is said to be an '''atom (of $\mu$)''' : :$(1): \quad \set x \in \Sigma$ :$(2): \quad \map \mu {\set x} > 0$
Definition:Standard Ordered Basis/Vector Space
Let $\struct {\mathbf V, +, \circ}_{\mathbb F}$ be a vector space over a field $\mathbb F$, as defined by the vector space axioms. Let the unity of $\mathbb F$ be denoted $1_{\mathbb F}$, and its zero $0_{\mathbb F}$. Let $\mathbf e_i$ be a vector whose $i$th term is $1_{\mathbb F}$ and with entries $0_{\mathbb F}$ elsewhere. Then the ordered $n$-tuple $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ is the '''standard ordered basis of $\mathbf V$'''.
Definition:Standard Basis/Vector Space
Let $\left({\mathbf V, +, \circ}\right)_{\mathbb F}$ be a vector space over $\mathbb F$. Let $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ be the standard ordered basis on $\mathbf V$. The corresponding (unordered) set $\left\{{\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right\}$ is called the '''standard basis of $\mathbf V$'''
Definition:Coordinate System
Let $R$ be a ring with unity. Let $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ be an ordered basis of a free $R$-module $G$. Then $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ can be referred to as a '''coordinate system'''.
Definition:Dimension of Module
Let $R$ be a ring with unity. Let $G$ be a free $R$-module which has a basis of $n$ elements. Then $G$ is said to have a '''dimension of $n$''' or to be '''$n$-dimensional'''. The dimension of a free $R$-module $G$ is denoted $\map {\dim_R} G$, or just $\map \dim G$.
Definition:Dimension of Vector Space
Let $K$ be a division ring. Let $V$ be a vector space over $K$. === Definition 1 === {{:Definition:Dimension of Vector Space/Definition 1}} === Definition 2 === {{:Definition:Dimension of Vector Space/Definition 2}}
Definition:Dimension of Module/Finite
Let $G$ be a (unitary) module which is $n$-dimensional for some $n \in \N_{>0}$. Then $G$ is '''finite dimensional'''.
Definition:Dimension of Vector Space/Finite
Let $V$ be a vector space which is $n$-dimensional for some $n \in \N_{>0}$. Then $V$ is '''finite dimensional'''. The '''dimension''' of a '''finite-dimensional $K$-vector space''' $V$ is denoted $\map {\dim_K} V$, or just $\map \dim V$.
Definition:Homomorphism (Abstract Algebra)/Image
As a homomorphism is a mapping, the '''homomorphic image''' of $\phi$ is defined in the same way as the image of a mapping: :$\Img \phi = \set {t \in T: \exists s \in S: t = \map \phi s}$
Definition:Linear Operator/Vector Space
A '''linear operator''' on a vector space is a linear transformation from a vector space into itself.
Definition:Linear Transformation/Vector Space
Let $V, W$ be vector spaces over a field (or, more generally, division ring) $K$. A mapping $A: V \to W$ is a '''linear transformation''' : :$\forall v_1, v_2 \in V, \lambda \in K: \map A {\lambda v_1 + v_2} = \lambda \map A {v_1} + \map A {v_2}$ That is, a homomorphism from one vector space to another.
Definition:Kernel of Linear Transformation/Vector Space
Let $\struct {\mathbf V, +, \times}$ be a vector space. Let $\struct {\mathbf V', +, \times}$ be a vector space whose zero vector is $\mathbf 0'$. Let $T: \mathbf V \to \mathbf V'$ be a linear transformation. Then the '''kernel''' of $T$ is defined as: :$\map \ker T := T^{-1} \sqbrk {\set {\mathbf 0'} } = \set {\mathbf x \in \mathbf V: \map T {\mathbf x} = \mathbf 0'}$
Definition:Rank/Linear Transformation
Let $\phi$ be a linear transformation from one vector space to another. Let the image of $\phi$ be finite-dimensional. Then its dimension is called the '''rank of $\phi$''' and is denoted $\map \rho \phi$.
Definition:Rank/Matrix
=== Definition 1 === {{:Definition:Rank/Matrix/Definition 1}} === Definition 2 === {{:Definition:Rank/Matrix/Definition 2}} === Definition 3 === {{:Definition:Rank/Matrix/Definition 3}}
Definition:Nullity/Linear Transformation
Let $K$ be a division ring. Let $V$ and $W$ be $K$-vector spaces. Let $\phi: V \to W$ be a linear transformation. Let the kernel $\ker \phi$ be finite dimensional. Then the '''nullity of $\phi$''' is the dimension of $\ker \phi$ and is denoted $\map \nu \phi$.
Definition:Nullity/Matrix
Let $\mathbf A$ be a matrix. Then the '''nullity of $\mathbf A$''' is defined to be the dimension of the null space of $\mathbf A$.
Definition:Vector Notation
Several conventions are found in the literature for annotating a general vector quantity in a style that distinguishes it from a scalar quantity, as follows. Let $\set {x_1, x_2, \ldots, x_n}$ be a collection of scalars which form the components of an $n$-dimensional vector. The vector $\tuple {x_1, x_2, \ldots, x_n}$ can be annotated as: {{begin-eqn}} {{eqn | l = \bsx       | r = \tuple {x_1, x_2, \ldots, x_n} }} {{eqn | l = \vec x       | r = \tuple {x_1, x_2, \ldots, x_n} }} {{eqn | l = \hat x       | r = \tuple {x_1, x_2, \ldots, x_n} }} {{eqn | l = \underline x       | r = \tuple {x_1, x_2, \ldots, x_n} }} {{eqn | l = \tilde x       | r = \tuple {x_1, x_2, \ldots, x_n} }} {{end-eqn}} To emphasize the arrow interpretation of a vector, we can write: :$\bsv = \sqbrk {x_1, x_2, \ldots, x_n}$ or: :$\bsv = \sequence {x_1, x_2, \ldots, x_n}$ In printed material the '''boldface''' $\bsx$ or $\mathbf x$ is common. This is the style encouraged and endorsed by . However, for handwritten material (where boldface is difficult to render) it is usual to use the '''underline''' version $\underline x$.  Also found in handwritten work are the '''tilde''' version $\tilde x$ and '''arrow''' version $\vec x$, but as these are more intricate than the simple underline (and therefore more time-consuming and tedious to write), they will only usually be found in fair copy. It is also noted that the '''tilde''' over $\tilde x$ does not render well in  under all browsers, and differs little visually from an '''overline''': $\overline x$. The '''hat''' version $\hat x$ usually has a more specialized meaning, namely to symbolize a unit vector. In computer-rendered materials, the '''arrow''' version $\vec x$ is popular, as it is descriptive and relatively unambiguous, and in $\LaTeX$ it is straightforward. However, it does not render well in all browsers, and is therefore (reluctantly) not recommended for use on this website.
Definition:Annihilator on Algebraic Dual
Let $R$ be a commutative ring. Let $G$ be a module over $R$. Let $G^*$ be the algebraic dual of $G$. Let $M$ be a submodule of $G$. The '''annihilator of $M$''', denoted $M^\circ$, is defined as: :$\ M^\circ := \set {t' \in G^*: \forall x \in M: \map {t'} x = 0}$
Definition:Matrix/Order
Let $\sqbrk a_{m n}$ be an $m \times n$ matrix. Then the parameters $m$ and $n$ are known as the '''order''' of the matrix.
Definition:Matrix/Row
Let $\mathbf A$ be an $m \times n$ matrix. For each $i \in \closedint 1 m$, the '''rows''' of $\mathbf A$ are the ordered $n$-tuples: :$r_i = \tuple {a_{i 1}, a_{i 2}, \ldots, a_{i n} }$ where $r_i$ is called the '''$i$th row of $\mathbf A$'''. A '''row''' of an $m \times n$ matrix can also be treated as a $1 \times n$ row matrix in its own right: :$r_i = \begin {bmatrix} a_{i 1} & a_{i 2} & \cdots & a_{i n} \end {bmatrix}$ for $i = 1, 2, \ldots, m$.