hcju/proofwikips huggingface dataset:
Definition:Minimal Hausdorff Topology
:thumbright420pxExamples of [[Definition:Neighborhood Basisneighborhood bases of $a$ and $-a$]] Let $\omega$ be the first transfinite ordinal. Let $\struct {A, \tau_1}$ denote the totally ordered set defined as: :$A := \set {1, 2, 3, \ldots, \omega, \ldots, -3, -2, -1}$ with the interval topology $\tau_1$. Let $\struct {\Z_{>0}, \tau_2}$ denote the (strictly) positive integers with the discrete topology $\tau_2$. Let $S$ be defined as: :$S = A \times \Z_{>0} \cup \set {a, -a}$ where $a$ and $-a$ are new elements of $S$. Let $\tau$ be the topology defined on $S$ as: :the product topology on $\struct {A, \tau_1} \times \struct {\Z_{>0}, \tau_2}$ together with neighborhood bases of $a$ and $-a$ defined as: {{begin-eqn}} {{eqn | l = \map { {M_n}^+} a       | o = :=       | r = \set a \cup \set {\tuple {i, j}: i < \omega, j > n} }} {{eqn | l = \map { {M_n}^-} {-a}       | o = :=       | r = \set {-a} \cup \set {\tuple {i, j}: i > \omega, j > n} }} {{end-eqn}} $\tau$ is referred to as the '''minimal Hausdorff topology'''. The topological space $T = \struct {S, \tau}$ is referred to as the '''minimal Hausdorff space'''.
Definition:Complex Vector Space
Let $\C$ be the set of complex numbers. Then the $\C$-module $\C^n$ is called the '''complex ($n$-dimensional) vector space'''.
Definition:Linearly Independent/Set/Complex Vector Space
Let $\tuple {\C^n, +, \cdot}_\C$ be a complex vector space. Let $S \subseteq \C^n$. Then $S$ is a '''linearly independent set of complex vectors''' if every finite sequence of distinct terms in $S$ is a linearly independent sequence. That is, such that: :$\displaystyle \forall \set {\lambda_k: 1 \le k \le n} \subseteq \C: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0 \implies \lambda_1 = \lambda_2 = \cdots = \lambda_n = 0$ where $\mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n$ are distinct elements of $S$.
Definition:Linearly Dependent/Set/Complex Vector Space
Let $\struct {\C^n, +, \cdot}_\C$ be a complex vector space. Let $S \subseteq \C^n$. 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: :$\displaystyle \exists \set {\lambda_k: 1 \le k \le n} \subseteq \C: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0$ 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:Pointwise Operation/Induced Structure
The algebraic structure $\struct {T^S, \oplus}$ is called the '''algebraic structure on $T^S$ induced by $\circ$'''.
Definition:Pointwise Addition of Linear Operators
Let $V$ be a vector space. Let $\map \LL V$ denote the set of linear operators on $V$. :$+: \map \LL V \times \map \LL V \to \map \LL V: \forall S, T \in \map \LL V:$ ::$\forall u \in V: \map {\paren {S + T} } u := \map S u + \map T u$ where $+$ on the  is vector addition.
Definition:Pointwise Addition on Complex Vector Space
Let $\C^n$ be a complex vector space. Let $S$ and $T$ be linear operators on $\C^n$. Then the '''pointwise sum of $S$ and $T$''' is defined as: :$S + T: \C^n \to \C^n:$ ::$\forall u \in \C^n: \map {\paren {S + T} } u := \map S u + \map T u$ where $+$ on the  is complex vector addition.
Definition:Matrix Space/Real
Let $\R$ denote the set of real numbers. The '''$m \times n$ matrix space over $\R$''' is referred to as the '''real matrix space''', and can be denoted $\map {\MM_\R} {m, n}$.
Definition:Matrix Space/Complex
Let $\C$ denote the set of complex numbers. The '''$m \times n$ matrix space over $\C$''' is referred to as the '''complex matrix space''', and can be denoted $\map {\MM_\C} {m, n}$.
Definition:Ones Matrix/Square
A '''square ones matrix of order $n$''' is a square matrix of order $n$ all of whose elements are $1$. It is often denoted $\mathbf J_n$.
Definition:Matrix Entrywise Addition/Ring
Let $\struct {R, +, \cdot}$ be a ring. Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $R$. Let $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$. Then the '''matrix entrywise sum of $\mathbf A$ and $\mathbf B$''' is written $\mathbf A + \mathbf B$, and is defined as follows. Let $\mathbf A + \mathbf B = \mathbf C = \sqbrk c_{m n}$. Then: :$\forall i \in \closedint 1 m, j \in \closedint 1 n: c_{i j} = a_{i j} + b_{i j}$ Thus $\sqbrk c_{m n}$ is the $m \times n$ matrix whose entries are made by performing the (ring) addition operation $+$ on corresponding entries of $\mathbf A$ and $\mathbf B$.
Definition:Limit of Sequence/Normed Vector Space
Let $L \in X$. Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$. Let $\sequence {x_n}_{n \mathop \in \N}$ converge to $L$.  Then $L$ is a '''limit of  $\sequence {x_n}_{n \mathop \in \N}$ as $n$ tends to infinity''' which is usually written: :$\displaystyle L = \lim_{n \mathop \to \infty} x_n$
Definition:Zero Matrix/Ring
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. Let $\map {\MM_R} {m, n}$ be an $m \times n$ matrix space over $R$. The '''zero matrix of $\map {\MM_R} {m, n}$''', denoted $\mathbf 0_R$, is the $m \times n$ matrix whose elements are all $0_R$, and can be written $\sqbrk {0_R}_{m n}$.
Definition:Complex Matrix
A '''complex matrix''' is a matrix whose elements are all complex numbers.
Definition:Zero Matrix
Let $\Bbb F$ be one of the standard number system $\N$, $\Z$, $\Q$, $\R$ and $\C$. Let $\map \MM {m, n}$ be an $m \times n$ matrix space over $\Bbb F$. The '''zero matrix of $\map \MM {m, n}$''', denoted $\mathbf 0$, is the $m \times n$ matrix whose elements are all zero, and can be written $\sqbrk 0_{m n}$.
Definition:Matrix Scalar Product/Ring
Let $\struct {R, +, \circ}$ be a ring. Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over $\struct {R, +, \circ}$. Let $\lambda \in R$ be any element of $R$. The '''scalar product of $\lambda$ and $\mathbf A$''' is defined as follows. Let $\lambda \circ \mathbf A = \mathbf C$. Then: :$\forall i \in \closedint 1 m, j \in \closedint 1 n: c_{i j} = \lambda \circ a_{i j}$ Thus $\sqbrk c_{m n}$ is the $m \times n$ matrix composed of the product of $\lambda$ with the corresponding elements of $\mathbf A$.
Definition:Negative Matrix/Ring
Let $\struct {R, +, \circ}$ be a ring. Let $\map {\MM_R} {m, n}$ denote the $m \times n$ matrix space over $\struct {R, +, \circ}$. Let $\mathbf A = \sqbrk a_{m n}$ be an element of $\map {\MM_R} {m, n}$. Then the '''negative (matrix) of $\mathbf A$''' is denoted and defined as: :$-\mathbf A := \sqbrk {-a}_{m n}$ where $-a$ is the ring negative of $a$.
Definition:Inverse Matrix/Right
Let $\mathbf A = \sqbrk a_{m n}$ be a matrix of order $m \times n$. Let $\mathbf B = \sqbrk b_{n m}$ be a matrix of order $n \times m$ such that: :$\mathbf A \mathbf B = I_m$ where $I_m$ denotes the unit matrix of order $m$. Then $\mathbf B$ is known as a '''right inverse (matrix)''' of $\mathbf A$.
Definition:Inverse Matrix/Left
Let $\mathbf A = \sqbrk a_{m n}$ be a matrix of order $m \times n$. Let $\mathbf B = \sqbrk b_{n m}$ be a matrix of order $n \times m$ such that: :$\mathbf B \mathbf A = I_n$ where $I_n$ denotes the unit matrix of order $n$. Then $\mathbf B$ is known as a '''left inverse (matrix)''' of $\mathbf A$.
Definition:Row Operation
Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over a field $K$. A '''row operation''' on $\mathbf A$ is a sequence of '''elementary row operations''' performed on $\mathbf A$ in turn.
Definition:Echelon Matrix/Echelon Form/Non-Unity Variant/Definition 1
$\mathbf A$ is in '''non-unity echelon form''' : :$(1): \quad$ Each row (except perhaps row $1$) starts with a sequence of zeroes :$(2): \quad$ Except when for row $k$ and row $k + 1$ are zero rows, the number of zeroes in this initial sequence in row $k + 1$ is strictly greater than the number of zeroes in this initial sequence in row $k$ :$(3): \quad$ The non-zero rows appear before any zero rows.
Definition:Echelon Matrix/Echelon Form/Non-Unity Variant/Definition 2
$\mathbf A$ is in '''non-unity echelon form'''  it contains no adjacent rows of the form: :$\begin {pmatrix} 0 & 0 & \cdots & 0 & x_1 & x_2 & \cdots \\ 0 & 0 & \cdots & 0 & y_1 & y_2 & \cdots \\ \end {pmatrix}$ where: :$(1): \quad y_1 \ne 0$ :$(2): \quad x_1$ can be any value at all, including $0$.
Definition:Echelon Matrix/Echelon Form
$\mathbf A$ is in '''echelon form''' : :$(1): \quad$ The leading coefficient in each non-zero row is $1$ :$(2): \quad$ The leading $1$ in any non-zero row occurs to the right of the leading $1$ in any previous row :$(3): \quad$ The non-zero rows appear before any zero rows.
Definition:Transposition/Adjacent
Let $S_n$ denote the symmetric group on $n$ letters. An '''adjacent transposition''' is a transposition that exchanges two consecutive integers $j$ and $j + 1$, where $1 \le j < n$. In cycle notation, they are denoted: :$\begin {pmatrix} j & j + 1 \end {pmatrix}$
Definition:Rank/Matrix/Definition 1
Let $K$ be a field. Let $\mathbf A$ be an $m \times n$ matrix over $K$. Then the '''rank''' of $\mathbf A$, denoted $\map \rho {\mathbf A}$, is the dimension of the subspace of $K^m$ generated by the columns of $\mathbf A$. That is, it is the dimension of the column space of $\mathbf A$.
Definition:Gaussian Elimination
Let $\mathbf A$ be a matrix over a field $K$. Let $\mathbf E$ be a reduced echelon matrix which is row equivalent to $\mathbf A$. The '''Gaussian elimination method''' is a technique for converting $\mathbf A$ into $\mathbf E$ by means of a sequence of elementary row operations.
Definition:Simultaneous Linear Equations/Matrix Representation/Matrix of Coefficients
The matrix $\mathbf A$ is known as the '''matrix of coeffficients''' of the system.
Definition:Taxicab Norm
Let $\mathbf v = \tuple {v_1, v_2, \ldots, v_n}$ be a vector in $\R^n$. The '''taxicab norm''' of $\mathbf v$ is defined as: :$\displaystyle \norm {\mathbf v}_1 = \sum_{k \mathop = 1}^n \size {v_k}$
Definition:Simultaneous Linear Equations/Matrix Representation/Augmented Matrix
Let $\begin {bmatrix} \mathbf A & \mathbf b \end {bmatrix}$ be the block matrix formed from $\mathbf A$ and $\mathbf b$. Then $\begin {bmatrix} \mathbf A & \mathbf b \end {bmatrix}$ is known as the '''augmented matrix''' of the system.
Definition:Augmented Matrix
Let $\mathbf A$ be a matrix of order $n \times m$. Let $\mathbf B$ be a matrix of order $n \times k$. The '''augmented matrix''' of $\mathbf A$ and $\mathbf B$ is the block matrix $\begin {pmatrix} \mathbf A & \mathbf B \end {pmatrix}$ of order $n \times \paren {m + k}$. === Augmented Matrix of Simultaneous Equations === {{:Definition:Simultaneous Linear Equations/Matrix Representation/Augmented Matrix}}
Definition:Simultaneous Equations/Linear Equations/Solution
Let $\tuple {x_1, x_2, \ldots, x_n}$ satisfy each of the equations in $\displaystyle \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$. Then $\tuple {x_1, x_2, \ldots, x_n}$ is referred to as a '''solution to the '''system of simultaneous linear equations'''
Definition:Equivalent Systems of Simultaneous Linear Equations
Let $S_1$ and $S_2$ be two systems of simultaneous linear equations. Then $S_1$ and $S_2$ are '''equivalent''' : :every solution to $S_1$ is also a solution to $S_2$ and: :every solution to $S_2$ is also a solution to $S_1$.
Definition:Homogeneous Simultaneous Linear Equations
A '''system of simultaneous linear equations''': :$\displaystyle \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$ is referred to as '''homogeneous''' : :$\forall i \in \set {1, 2, \ldots, m}: \beta_i = 0$ That is: {{begin-eqn}} {{eqn | r = 0       | l = \alpha_{1 1} x_1 + \alpha_{1 2} x_2 + \cdots + \alpha_{1 n} x_n }} {{eqn | r = 0       | l = \alpha_{2 1} x_1 + \alpha_{2 2} x_2 + \cdots + \alpha_{2 n} x_n }} {{eqn | o = \cdots}} {{eqn | r = 0       | l = \alpha_{m 1} x_1 + \alpha_{m 2} x_2 + \cdots + \alpha_{m n} x_n }} {{end-eqn}}
Definition:Consistent Simultaneous Linear Equations
A '''system of simultaneous linear equations''': :$\displaystyle \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$ is referred to as '''consistent'''  it has at least one solution.
Definition:Trivial Solution to Homogeneous Simultaneous Linear Equations
Let $S$ be a '''system of homogeneous simultaneous linear equations''': :$\displaystyle \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = 0$ The solution: :$\tuple {x_1, x_2, \ldots, x_n}$ such that: :$\forall j \in \set {1, 2, \ldots, n}: x_j = 0$ is known as the '''trivial solution''' to $S$.
Definition:Scalar
=== R-Algebraic Structure === Let $\struct {R, +_R, \times_R}$ be the scalar ring of an $R$-algebraic structure $\struct {S, *_1, *_2, \ldots, *_n, \circ}_R$. {{:Definition:Scalar/R-Algebraic Structure}} === Module === Let $\struct {R, +_R, \times_R}$ be the scalar ring of a module $\struct {G, +_G, \circ}_R$. {{:Definition:Scalar/Module}} === Vector Space === Let $\struct {K, +_K, \times_K}$ be the scalar field of a vector space $\struct {G, +_G, \circ}_K$. {{:Definition:Scalar/Vector Space}} === Scalar (Matrix Theory) === {{:Definition:Scalar (Matrix Theory)}} === Scalar Quantity === {{:Definition:Scalar Quantity}}
Definition:Scalar (Matrix Theory)
Let $\map \MM {m, n}$ be a matrix space of order $m \times n$ over an underlying structure $R$. The elements of $R$ are referred to as '''scalars'''.
Definition:Matrix Scalar Product/Scalar
The element $\lambda$ of the underlying structure of $\map \MM {m, n}$ is known as a '''scalar'''.
Definition:Equality of Matrices
Let $\mathbf A$ and $\mathbf B$ be matrices over an underlying structure $R$. Then $\mathbf A$ is equal to $\mathbf B$ : :$(1): \quad$ the order of $\mathbf A$ equals the order of $\mathbf B$: $m \times n$, say :$(2): \quad$ for all $i \in \set {1, 2, \ldots, m}$ and $j \in \set {1, 2, \ldots, n}$, we have that $a_{i j} = b_{i j}$.
Definition:Matrix Entrywise Addition/General
Let $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_k$ be matrices all of order of $m \times n$. Then the '''matrix entrywise sum of $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_k$''' is written $\mathbf A_1 + \mathbf A_2 + \ldots + \mathbf A_k$, and is defined as follows: Then: :$\forall i \in \closedint 1 m, j \in \closedint 1 n: K_{i j} = \paren {a_1}_{i j} + \paren {a_2}_{i j} + \cdots + \paren {a_k}_{i j}$ where $\paren {a_l}_{i j}$ is the element of $\mathbf A_l$ whose indices are $\tuple {i, j}$. Thus $\mathbf K = \sqbrk k_{m n}$ is the $m \times n$ matrix whose entries are made by performing the adding corresponding entries of $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_k$. This can be expressed in summation form as: :$\displaystyle K = \sum_{j \mathop = 1}^k \mathbf A_j$
Definition:Rank/Matrix/Definition 2
Let $K$ be a field. Let $\mathbf A$ be an $m \times n$ matrix over $K$. Let $\mathbf A$ be converted to echelon form $\mathbf B$. Let $\mathbf B$ have exactly $k$ non-zero rows. Then the '''rank''' of $\mathbf A$, denoted $\map \rho {\mathbf A}$, is $k$.
Definition:Perfect Field/Definition 1
$F$ is a '''perfect field '''  $F$ has no inseparable extensions.
Definition:Perfect Field/Definition 2
$F$ is a '''perfect field'''  one of the following holds: :$\Char F = 0$ :$\Char F = p$ with $p$ prime and $\Frob$ is an automorphism of $F$ where: :$\Char F$ denotes the characteristic of $F$ :$\Frob$ denotes the Frobenius endomorphism on $F$
Definition:Elementary Operation/Column
Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over a field $K$. The '''elementary column operations''' on $\mathbf A$ are operations which act upon the columns of $\mathbf A$ as follows. For some $i, j \in \closedint 1 n: i \ne j$: {{begin-axiom}} {{axiom | n = \text {ECO} 1         | t = For some $\lambda \in K_{\ne 0}$, multiply column $i$ by $\lambda$         | m = \kappa_i \to \lambda \kappa_i }} {{axiom | n = \text {ECO} 2         | t = For some $\lambda \in K$, add $\lambda$ times column $j$ to column $i$         | m = \kappa_i \to \kappa_i + \lambda \kappa_j }} {{axiom | n = \text {ECO} 3         | t = Interchange columns $i$ and $j$         | m = \kappa_i \leftrightarrow \kappa_j }} {{end-axiom}}
Definition:Column Operation
Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over a field $K$. A '''column operation''' on $\mathbf A$ is a sequence of '''elementary column operations''' performed on $\mathbf A$ in turn.
Definition:Column Equivalence
Two matrices $\mathbf A = \sqbrk a_{m n}, \mathbf B = \sqbrk b_{m n}$ are '''column equivalent''' if one can be obtained from the other by a finite sequence of elementary column operations. This relationship can be denoted $\mathbf A \sim \mathbf B$.
Definition:Determinant/Matrix/In Full
When written out in full, the determinant of $\mathbf A$ is denoted: :$\map \det {\mathbf A} = \begin {vmatrix} a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\  \vdots &  \vdots & \ddots &  \vdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n} \\ \end {vmatrix}$
Definition:Determinant/Matrix/Definition 1
Let $\lambda: \N_{> 0} \to \N_{> 0}$ be a permutation on $\N_{>0}$. The '''determinant''' of $\mathbf A$ is defined as: :$\displaystyle \map \det {\mathbf A} := \sum_{\lambda} \paren {\map \sgn \lambda \prod_{k \mathop = 1}^n a_{k \map \lambda k} } = \sum_\lambda \map \sgn \lambda a_{1 \map \lambda 1} a_{2 \map \lambda 2} \cdots a_{n \map \lambda n}$ 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 $\lambda$.
Definition:Determinant/Matrix/Definition 2
The '''determinant''' of $\mathbf A$ is defined as follows: For $n = 1$, the order $1$ determinant is defined as: {{:Definition:Determinant/Matrix/Order 1}} For $n > 1$, the determinant of order $n$ is defined recursively as: :$\displaystyle \map \det {\mathbf A} := \begin {vmatrix} a_{1 1} & a_{1 2} & a_{1 3} & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & a_{2 3} & \cdots & a_{2 n} \\ a_{3 1} & a_{3 2} & a_{3 3} & \cdots & a_{3 n} \\  \vdots &  \vdots &  \vdots & \ddots &  \vdots \\ a_{n 1} & a_{n 2} & a_{n 3} & \cdots & a_{n n} \\ \end {vmatrix} = a_{1 1} \begin {vmatrix} a_{2 2} & a_{2 3} & \cdots & a_{2 n} \\ a_{3 2} & a_{3 3} & \cdots & a_{3 n} \\  \vdots &  \vdots & \ddots &  \vdots \\ a_{n 2} & a_{n 3} & \cdots & a_{n n} \\ \end {vmatrix} - a_{1 2} \begin {vmatrix} a_{2 1} & a_{2 3} & \cdots & a_{2 n} \\ a_{3 1} & a_{3 3} & \cdots & a_{3 n} \\  \vdots &  \vdots & \ddots &  \vdots \\ a_{n 1} & a_{n 3} & \cdots & a_{n n} \\ \end {vmatrix} + \cdots + \paren {-1}^{n + 1} a_{1 n} \begin {vmatrix} a_{2 1} & a_{2 2} & \cdots & a_{2, n - 1} \\ a_{3 1} & a_{3 3} & \cdots & a_{3, n - 1} \\  \vdots &  \vdots & \ddots &       \vdots \\ a_{n 1} & a_{n 3} & \cdots & a_{n, n - 1} \\ \end {vmatrix}$
Definition:Polygamma Function
The $n$th '''polygamma function''', $\psi_n$, is defined, for $z \in \C \setminus \Z_{\le 0}$, by the $n$th derivative of the digamma function:  :$\map {\psi_n} z = \dfrac {\d^n} {\d z^n} \map \psi z$ where $\psi$ is the digamma function.
Definition:Elementary Matrix/Row Operation
Let $\mathbf E$ be a unit matrix on which exactly one elementary row operation $e$ has been performed. Then $\mathbf E$ is called the '''elementary row matrix''' for the elementary row operation $e$.
Definition:Elementary Matrix/Column Operation
Let $\mathbf E$ be a unit matrix on which exactly one elementary column operation $e$ has been performed. Then $\mathbf E$ is called the '''elementary column matrix''' for the elementary column operation $e$.
Definition:Elementary Matrix Operation
=== Elementary Row Operation === {{:Definition:Elementary Row Operation}} === Elementary Column Operation === {{:Definition:Elementary Column Operation}}
Definition:Elementary Operation/Row
Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over a field $K$. The '''elementary row operations''' on $\mathbf A$ are operations which act upon the rows of $\mathbf A$ as follows. For some $i, j \in \closedint 1 m: i \ne j$: {{begin-axiom}} {{axiom | n = \text {ERO} 1         | t = For some $\lambda \in K_{\ne 0}$, multiply row $i$ by $\lambda$         | m = r_i \to \lambda r_i }} {{axiom | n = \text {ERO} 2         | t = For some $\lambda \in K$, 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}}
Definition:Matrix Product (Conventional)/Pre-Multiplication
Let $\mathbf A \mathbf B$ be the product of $\mathbf A$ with $\mathbf B$. Then $\mathbf B$ is '''pre-multiplied''' by $\mathbf A$.
Definition:Matrix Product (Conventional)/Post-Multiplication
Let $\mathbf A \mathbf B$ be the product of $\mathbf A$ with $\mathbf B$. Then $\mathbf A$ is '''post-multiplied''' by $\mathbf B$.
Definition:Distance/Points/Normed Vector Space
Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $x, y \in X$ . Then the function $\norm {\, \cdot \,} : X \times X \to \R$: :$\map d {x, y} = \norm {x - y}$ is called the '''distance between $x$ and $y$'''.
Definition:Inverse of Elementary Row Operation
Let $e$ be an elementary row operation which transforms a matrix $\mathbf A$ to another matrix $\mathbf B$. Let $e'$ be an elementary row operation which transforms $\mathbf B$ back to $\mathbf A$. Then $e'$ is the '''inverse of the elementary row operation $e$'''.
Definition:Inverse of Elementary Column Operation
Let $e$ be an elementary column operation which transforms a matrix $\mathbf A$ to another matrix $\mathbf B$. Let $e'$ be an elementary column operation which transforms $\mathbf B$ back to $\mathbf A$. Then $e'$ is the '''inverse of the elementary column operation $e$'''.
Definition:Extended Weight Function
Let $S$ be a set. Let $\mathscr F$ be the set of all finite subsets of $S$. Let $w: S \to \R$ be a weight function. The '''extended weight function''' of $w$ is the function $w^+: \mathscr F \to \R$ defined by: :$\forall A \in \mathscr F : \map {w^+} A = \displaystyle \sum_{a \mathop \in A} \map w a$
Definition:Matrix of Minors
Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. Let $a_{i j}$ denote the element whose indices are $\tuple {i, j}$: :$\mathbf A = \begin {pmatrix} a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\  \vdots &  \vdots & \ddots &  \vdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n} \\ \end {pmatrix}$ For each $a_{i j}$, let $b_{i j}$ denote the minor of $a_{i j}$: :$b_{i j} = \begin {vmatrix} a_{1 1} & a_{1 2} & \cdots & a_{1, i - 1} & a_{1, i + 1} & \cdots & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & \cdots & a_{2, i - 1} & a_{2, i + 1} & \cdots & \cdots & a_{2 n} \\  \vdots &  \vdots & \ddots & \vdots &  \vdots & \ddots & \vdots \\ a_{j - 1, 1} & a_{j - 1, 2} & \cdots & a_{j - 1, i - 1} & a_{j - 1, i + 1} & \cdots & \cdots & a_{j - 1, n} \\ a_{j + 1, 1} & a_{j + 1, 2} & \cdots & a_{j + 1, i - 1} & a_{j + 1, i + 1} & \cdots & \cdots & a_{j + 1, n} \\  \vdots &  \vdots & \ddots & \vdots &  \vdots & \ddots & \vdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n, i - 1} & a_{n, i + 1} & \cdots & \cdots & a_{n n} \\ \end {vmatrix}$ Then $\mathbf B = \sqbrk b_n$ is called the '''matrix of minors''' of $\mathbf A$. :$\mathbf B = \begin {pmatrix} b_{1 1} & b_{1 2} & \cdots & b_{1 n} \\ b_{2 1} & b_{2 2} & \cdots & b_{2 n} \\  \vdots &  \vdots & \ddots &  \vdots \\ b_{n 1} & b_{n 2} & \cdots & b_{n n} \\ \end {pmatrix}$