nyuuzyou/wikis huggingface dataset:
Elementary Properties of Event Space

== Theorem ==
Let $\EE$ be an experiment with a probability space $\struct {\Omega, \Sigma, \Pr}$.
The event space $\Sigma$ of $\EE$ has the following properties:
=== Event Space contains Empty Set ===
=== Event Space contains Sample Space ===
=== Intersection of Events is Event ===
=== Set Difference of Events is Event ===
=== Symmetric Difference of Events is Event ===
=== Countable Intersection of Events is Event ===
In the above:
:$A \setminus B$ denotes set difference
:$A \symdif B$ denotes symmetric difference.
Category:Event Spaces
Elementary Properties of Probability Measure

== Theorem ==
Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$.
The probability measure $\Pr$ of $\EE$ has the following properties:
=== Probability of Empty Event is Zero ===
=== Probability of Event not Occurring ===
=== Probability Measure is Monotone ===
Category:Probability Theory
Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse

== Theorem ==
Let $e$ be an elementary row operation.
Let $\mathbf E$ be the elementary row matrix corresponding to $e$.
Let $e'$ be the inverse of $e$.
Then the elementary row matrix corresponding to $e'$ is the inverse of $\mathbf E$.
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse
Category:Elementary Row Operations
Category:Elementary Matrices
Category:Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse
Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse/Proof 1

== Theorem ==
== Proof ==
We will demonstrate this for each of the $3$ types of elementary row operation.
Let $\sim$ denote row equivalence.
Let $\mathbf I$ denote the unit matrix.
=== $\text {ERO} 1$: Scalar Product of Row ===
Let $e_1$ be the elementary row operation:
:$e_1 := r_i \to \lambda r_i$
for an arbitrary row $r_i$ of $\mathbf I$.
Let $\mathbf E_1$ be the elementary row matrix created by applying $e_1$ to $\mathbf I$.
From Existence of Inverse Elementary Row Operation: Scalar Product of Row, the inverse $e_1'$ of $e_1$ is given by:
:$e_1' := r_i \to \dfrac 1 \lambda r_i$
Thus applying $e_1'$ to $\mathbf E_1$ transforms $\mathbf E_1$ back to $\mathbf I$.
From Elementary Row Operations as Matrix Multiplications, for every elementary row operation there exists a corresponding elementary row matrix. 
Let $\mathbf E_1'$ be the elementary row matrix created by applying $e_1'$ to $\mathbf I$.
Then:
:$\mathbf E_1 \mathbf E_1' = \mathbf E_1' \mathbf E_1 = \mathbf I$
=== $\text {ERO} 2$: Add Scalar Product of Row to Another ===
Let $e_2$ be the elementary row operation:
:$e_2 := r_i \to r_i + \lambda r_j$
for arbitrary row $r_i$ and $r_j$ of $\mathbf I$ such that $i \ne j$.
Let $\mathbf E_2$ be the elementary row matrix created by applying $e_2$ to $\mathbf I$.
From Existence of Inverse Elementary Row Operation: Add Scalar Product of Row to Another, the inverse $e_2'$ of $e_2$ is given by:
:$e_2' := r_i \to r_i - \lambda r_j$
Thus applying $e_2'$ to $\mathbf E_2$ transforms $\mathbf E_2$ back to $\mathbf I$.
From Elementary Row Operations as Matrix Multiplications, for every elementary row operation there exists a corresponding elementary row matrix. 
Let $\mathbf E_2'$ be the elementary row matrix created by applying $e_2'$ to $\mathbf I$.
Then:
:$\mathbf E_2 \mathbf E_2' = \mathbf E_2' \mathbf E_2 = \mathbf I$
=== $\text {ERO} 3$: Exchange Rows ===
Let $e_3$ be the elementary row operation:
:$e_3 := r_i \leftrightarrow r_j$
for arbitrary row $r_i$ and $r_j$ of $\mathbf I$ such that $i \ne j$.
Let $\mathbf E_3$ be the elementary row matrix created by applying $e_3$ to $\mathbf I$.
From Existence of Inverse Elementary Row Operation: Exchange Rows, the inverse $e_3'$ of $e_3$ is $e_3$ itself:
:$e_3' := r_i \leftrightarrow r_j = e_3$
Thus applying $e_3$ to $\mathbf E_3$ transforms $\mathbf E_3$ back to $\mathbf I$.
:$\mathbf E_3 \mathbf E_3 = \mathbf I$
Hence the result, from Proof by Cases.
== Sources ==
* $\S 1.5$
Category:Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse
Elementary Row Matrix is Nonsingular

== Theorem ==
Let $\mathbf E$ be an elementary row matrix.
Then $\mathbf E$ is nonsingular.
== Proof ==
From Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse it is demonstrated that:
:if $\mathbf E$ is the elementary row matrix corresponding to an elementary row operation $e$
then:
:the inverse of $e$ corresponds to an elementary row matrix which is the inverse of $\mathbf E$.
So as $\mathbf E$ has an inverse, it is nonsingular.
== Also see ==
* Elementary Column Matrix is Nonsingular
== Sources ==
* $\S 1.5$
Category:Elementary Matrices
Category:Nonsingular Matrices
Elementary Row Operation/Examples

== Examples of Elementary Row Operations ==
=== Example: $r_2 \to \lambda r_2$ ===
=== Example: $r_3 \to r_3 + 2 r_2$ ===
=== Example: $r_1 \leftrightarrow r_2$ ===
=== Arbitrary Operation on Identity Matrix ===
== Operations on Arbitrary Matrix ==
Category:Examples of Elementary Row Operations
Elementary Row Operation/Examples/Arbitrary Operation on Identity 1

== Example of Elementary Row Operation ==
The matrix:
:$\mathbf A = \begin {pmatrix} 1 & 0 \\ 1 & 1 \end {pmatrix}$
can be obtained from the identity matrix $\mathbf I_2$ by the elementary row operation $e$ defined as:
:$e := r_2 \to r_1 + r_2$
Then multiplying the matrix:
:$\mathbf X = \begin {pmatrix} a & b \\ c & d \end {pmatrix}$
on the left by $\mathbf A$
we get:
:$\begin {pmatrix} a & b \\ a + c & b + d \end {pmatrix}$
which can be obtained by applying that same elementary row operation $e$ on $\mathbf X$.
== Sources ==
* 
Category:Examples of Elementary Row Operations
Elementary Row Operation/Examples/Operations on Arbitrary Matrix

== Examples of Elementary Row Operations ==
Let $\mathbf A$ be the matrix:
:$\mathbf A = \begin {pmatrix} 1 & 2 & 3 & 4 \\ 2 & -1 & 1 & 0 \\ -2 & 3 & 1 & 1 \end {pmatrix}$
=== Example: $r_2 \to \lambda r_2$ ===
=== Example: $r_3 \to r_3 + 2 r_2$ ===
=== Example: $r_1 \leftrightarrow r_2$ ===
Category:Examples of Elementary Row Operations
Elementary Row Operation/Examples/Operations on Arbitrary Matrix/Swap r1 and r2

== Example of Elementary Row Operation ==
Let $\mathbf A$ be the matrix:
:$\mathbf A = \begin {pmatrix} 1 & 2 & 3 & 4 \\ 2 & -1 & 1 & 0 \\ -2 & 3 & 1 & 1 \end {pmatrix}$
Let the elementary row operation $e$ be applied to $\mathbf A$, where $e$ is defined as:
:$e := r_1 \leftrightarrow r_2$
Then $\mathbf A$ is transformed into:
:$\mathbf A = \begin {pmatrix} 2 & -1 & 1 & 0 \\ 1 & 2 & 3 & 4 \\ -2 & 3 & 1 & 1 \end {pmatrix}$
== Proof ==
From Elementary Row Operation: $r_1 \leftrightarrow r_2$, the elementary row matrix $\mathbf E$ corresponding to $e$ is:
:$\mathbf E = \begin {pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end {pmatrix}$
Hence by Elementary Row Operations as Matrix Multiplications:
== Sources ==
* : Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations: Exercise $1.1$
Category:Examples of Elementary Row Operations
Elementary Row Operation/Examples/Operations on Arbitrary Matrix/lambda r2

== Example of Elementary Row Operation ==
Let $\mathbf A$ be the matrix:
:$\mathbf A = \begin {pmatrix} 1 & 2 & 3 & 4 \\ 2 & -1 & 1 & 0 \\ -2 & 3 & 1 & 1 \end {pmatrix}$
Let the elementary row operation $e$ be applied to $\mathbf A$, where $e$ is defined as:
:$e := r_2 \to \lambda r_2$
Then $\mathbf A$ is transformed into:
:$\mathbf A = \begin {pmatrix} 1 & 2 & 3 & 4 \\ 2 \lambda & -\lambda & \lambda & 0 \\ -2 & 3 & 1 & 1 \end {pmatrix}$
== Proof ==
From Elementary Row Operation: $r_2 \to \lambda r_2$, the elementary row matrix $\mathbf E$ corresponding to $e$ is:
:$\mathbf E = \begin {pmatrix} 1 & 0 & 0 \\ 0 & \lambda & 0 \\ 0 & 0 & 1 \end {pmatrix}$
Hence by Elementary Row Operations as Matrix Multiplications:
== Sources ==
* : Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations: Exercise $1.1$
Category:Examples of Elementary Row Operations
Elementary Row Operation/Examples/Operations on Arbitrary Matrix/r3 + 2r2

== Example of Elementary Row Operation ==
Let $\mathbf A$ be the matrix:
:$\mathbf A = \begin {pmatrix} 1 & 2 & 3 & 4 \\ 2 & -1 & 1 & 0 \\ -2 & 3 & 1 & 1 \end {pmatrix}$
Let the elementary row operation $e$ be applied to $\mathbf A$, where $e$ is defined as:
:$e := r_3 \to r_3 + 2 r_2$
Then $\mathbf A$ is transformed into:
:$\mathbf A = \begin {pmatrix} 1 & 2 & 3 & 4 \\ 2 & -1 & 1 & 0 \\ 2 & 1 & 3 & 1 \end {pmatrix}$
== Proof ==
From Elementary Row Operation: $r_3 \to r_3 + 2 r_2$, the elementary row matrix $\mathbf E$ corresponding to $e$ is:
:$\mathbf E = \begin {pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 2 & 1 \end {pmatrix}$
Hence by Elementary Row Operations as Matrix Multiplications:
== Sources ==
* : Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations: Exercise $1.1$
Category:Examples of Elementary Row Operations
Elementary Row Operation/Examples/Swap r1 and r2

== Examples of Elementary Row Operations ==
Consider the elementary row operation $e$ defined as:
:$e := r_1 \leftrightarrow r_2$
acting on a matrix space $\map \MM {3, n}$ for some $n \in \Z_{>0}$.
The elementary row matrix corresponding to $e$ is:
:$\begin {pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end {pmatrix}$
== Proof ==
By definition, the elementary row matrix corresponding to $e$ is found by applying $e$ to the unit matrix.
By definition of unit matrix:
:$\mathbf I = \begin {pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {pmatrix}$
Let $\mathbf E$ denote the elementary row matrix corresponding to $e$.
$\mathbf E$ is constructed by exchanging row $1$ with row $2$.
== Sources ==
* : Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations: $3$
Category:Examples of Elementary Row Operations
Elementary Row Operation/Examples/lambda r2

== Example of Elementary Row Operation ==
Consider the elementary row operation $e$ defined as:
:$e := r_2 \to \lambda r_2$
acting on a matrix space $\map \MM {3, n}$ for some $n \in \Z_{>0}$.
The elementary row matrix corresponding to $e$ is:
:$\begin {pmatrix} 1 & 0 & 0 \\ 0 & \lambda & 0 \\ 0 & 0 & 1 \end {pmatrix}$
== Proof ==
Let $\mathbf E$ denote the elementary row matrix corresponding to $e$.
From Elementary Matrix corresponding to Elementary Row Operation: Scale Row and Add:
:$E_{a b} = \begin {cases} \delta_{a b} & : a \ne 2 \\ \lambda \cdot \delta_{a b} & : a = 2 \end{cases}$
where:
:$E_{a b}$ denotes the element of $\mathbf E$ whose indices are $\tuple {a, b}$
:$\delta_{a b}$ is the Kronecker delta:
::$\delta_{a b} = \begin {cases} 1 & : \text {if $a = b$} \\ 0 & : \text {if $a \ne b$} \end {cases}$
That is:
:When $a \ne 2$, the elements of $\mathbf E$ are $0$ except for those on the main diagonal, when they are $1$
:When $a = 2$, the elements of $\mathbf E$ are $0$ except for those on the main diagonal, when they are $\lambda$.
Hence $\mathbf E$ can be constructed as described.
== Sources ==
* : Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations: $2$
Category:Examples of Elementary Row Operations
Elementary Row Operation/Examples/r3 + 2r2

== Examples of Elementary Row Operations ==
Consider the elementary row operation $e$ defined as:
:$e := r_3 \to r_3 + 2 r_2$
acting on a matrix space $\map \MM {3, n}$ for some $n \in \Z_{>0}$.
The elementary row matrix corresponding to $e$ is:
:$\begin {pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 2 & 1 \end {pmatrix}$
== Proof ==
Let $\mathbf E$ denote the elementary row matrix corresponding to $e$.
From Elementary Matrix corresponding to Elementary Row Operation: Scale Row and Add:
:$E_{a b} = \delta_{a b} + 2 \delta_{a 3} \cdot \delta_{2 b}$
where:
:$E_{a b}$ denotes the element of $\mathbf E$ whose indices are $\tuple {a, b}$
:$\delta_{a b}$ is the Kronecker delta:
::$\delta_{a b} = \begin {cases} 1 & : \text {if $a = b$} \\ 0 & : \text {if $a \ne b$} \end {cases}$
That is:
:When $a \ne 3$ and $b \ne 2$, elements of $\mathbf E$ are $0$ except for those on the main diagonal
:When $a = 3$ and $b = 2$, $E_{a b}$ equals $2 + \delta_{a b}$.
But as $a \ne b$ it follows that $\delta_{a b} = 0$.
Hence $\mathbf E$ can be constructed as described.
== Sources ==
* : Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations: $1$
Category:Examples of Elementary Row Operations
Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 1

== Corollary to Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations ==
Let $S$ be a system of simultaneous linear equations:
:$\ds \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$
Let $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ denote the augmented matrix of $S$.
Let $\begin {pmatrix} \mathbf R & \mathbf s \end {pmatrix}$ be a reduced echelon matrix derived from $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$.
Let $S'$ be the system of simultaneous linear equations:
:$\ds \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \rho_{i j} x_j = \sigma_i$
whose augmented matrix is $\begin {pmatrix} \mathbf R & \mathbf s \end {pmatrix}$.
Then $S$ and $S'$ are equivalent.
== Proof ==
By Matrix is Row Equivalent to Reduced Echelon Matrix, $\begin {pmatrix} \mathbf R & \mathbf s \end {pmatrix}$ can be obtained from $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ by means of a finite sequence of elementary row operations.
By Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations, the system of simultaneous linear equations whose augmented matrix is obtained from $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ in this way is equivalent to $S$ after each elementary row operation.
Hence the entire sequence of such systems of simultaneous linear equations are equivalent to $S$.
In particular, this applies to $S'$.
== Sources ==
* : Chapter $1$: Linear Equations and Matrices: $1.3$ Applications to Linear Equations: Corollary $1$
Category:Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations
Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 3

== Corollary to Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations ==
Let $S$ be a system of homogeneous simultaneous linear equations:
:$\ds \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = 0$
If $m < n$, then $S$ has at least one non-trivial solution.
== Proof ==
Let $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ denote the augmented matrix of $S$.
Because $S$ is homogeneous, we have that $\mathbf b = \mathbf 0$, and so its augmented matrix is $\begin {pmatrix} \mathbf A & \mathbf 0 \end {pmatrix}$.
Let $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$ be a reduced echelon matrix derived from $\begin {pmatrix} \mathbf A & \mathbf 0 \end {pmatrix}$.
Let $S'$ be the system of simultaneous linear equations:
:$\ds \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \rho_{i j} x_j = 0$
whose augmented matrix is $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$.
By Corollary 1, $S$ and $S'$ are equivalent.
Hence any every solution to $S'$ is also a solution to $S$.
Consider the structure of $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$.
Suppose the leading coefficients appear in columns which we name $j_1, j_2, \ldots, j_l$.
Let the remaining columns be named $j_{l + 1}, j_{l + 2}, \ldots, j_n$.
Then we have that $S'$ can be expressed as:
where $l + 1 \le m$.
Setting arbitrary values to $x_{j_k}$ for $l < k \le n$ gives us a non-trivial solution for $S$.
== Sources ==
* : Chapter $1$: Linear Equations and Matrices: $1.3$ Applications to Linear Equations: Corollary $3$
Category:Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations
Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 4

== Corollary to Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations ==
Let $S$ be a system of homogeneous simultaneous linear equations:
:$\ds \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = 0$
Let $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$ be a reduced echelon matrix derived from $\begin {pmatrix} \mathbf A & \mathbf 0 \end {pmatrix}$.
Let the number of non-zero rows of $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$ be $l$.
If $l = n$, then the only solution to $S$ is the trivial solution.
== Proof ==
Consider the structure of $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$.
Suppose the leading coefficients appear in columns which we name $j_1, j_2, \ldots, j_n$.
As there are $n$ columns as well as $n$ non-zero rows:
:each row has exactly one $1$ in it
:each column has exactly one $1$ in it.
Thus $S'$ can be expressed as:
and the result follows.
== Sources ==
* : Chapter $1$: Linear Equations and Matrices: $1.3$ Applications to Linear Equations: Corollary $3$
Category:Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations
Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 5

== Corollary to Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations ==
Let $S$ be a system of $n$ homogeneous simultaneous linear equations in $n$ variables:
:$\ds \forall i \in \set {1, 2, \ldots, n}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = 0$
$S$ has a non-trivial solution its reduced echelon matrix $R$ is not equal to the unit matrix $\mathbf I_n$.
== Proof ==
Let $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$ be a reduced echelon matrix derived from $\begin {pmatrix} \mathbf A & \mathbf 0 \end {pmatrix}$.
Let the number of non-zero rows of $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$ be $l$.
:$R = \mathbf I_n$.
Then $l = n$.
From Corollary 4, in this case the only solution to $S$ is the trivial solution.
Let $R \ne \mathbf I_n$.
As $R$ is a reduced echelon matrix, it must have at least one zero row.
Thus $l < n$.
From Corollary 3, in this case there is at least one non-trivial solution.
== Sources ==
* : Chapter $1$: Linear Equations and Matrices: $1.3$ Applications to Linear Equations: Corollary $4$
Category:Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations
Elementary Row Operations as Matrix Multiplications

== Theorem ==
Let $e$ be an elementary row operation.
Let $\mathbf E$ be the elementary row matrix of order $m$ defined as:
:$\mathbf E = \map e {\mathbf I}$
where $\mathbf I$ is the unit matrix.
Then for every $m \times n$ matrix $\mathbf A$:
:$\map e {\mathbf A} = \mathbf {E A}$
where $\mathbf {E A}$ denotes the conventional matrix product.
=== Corollary ===
== Proof ==
Let $s, t \in \closedint 1 m$ such that $s \ne t$.
=== Case $1$ ===
Let $e$ be the elementary row operation $r_s \to \lambda r_s$:
:$E_{ik} = \begin{cases}
\delta_{ik} & : i \ne s \\
\lambda \delta_{ik} & : i = s
\end{cases}$
where $\delta$ denotes the Kronecker delta.
Then:
=== Case $2$ ===
Let $e$ be the elementary row operation $r_s \to r_s + \lambda r_t$:
:$E_{i k} = \begin {cases}
\delta_{i k} & : i \ne s \\
\delta_{s k} + \lambda \delta_{t k} & : i = s
\end {cases}$
where $\delta$ denotes the Kronecker delta.
Then:
=== Case $3$ ===
Let $e$ be the elementary row operation $r_s \leftrightarrow r_t$:
By Exchange of Rows as Sequence of Other Elementary Row Operations, this elementary row operation can be expressed as:
:$\map {e_1 e_2 e_3 e_4} {\mathbf A} = \map e {\mathbf A}$
where the $e_i$ are elementary row operations of the other two types.
For each $e_i$, let $\mathbf E_i = \map {e_i} {\mathbf I}$.
Then:
== Also see ==
* Elementary Column Operations as Matrix Multiplications
== Sources ==
* 
* : Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations
* 
Category:Conventional Matrix Multiplication
Category:Elementary Row Operations
Category:Elementary Matrices
Elementary Row Operations as Matrix Multiplications/Corollary

== Theorem ==
Let $\mathbf X$ and $\mathbf Y$ be two $m \times n$ matrices that differ by exactly one elementary row operation.
Then there exists an elementary row matrix of order $m$ such that:
:$\mathbf {E X} = \mathbf Y$
== Proof ==
Let $e$ be the elementary row operation such that $e \paren {\mathbf X} = \mathbf Y$.
Then this result follows immediately from Elementary Row Operations as Matrix Multiplications:
:$e \paren {\mathbf X} = \mathbf {E X} = \mathbf Y$
where $\mathbf E = e \paren {\mathbf I}$.
== Sources ==
* $\S 1.4$
Category:Conventional Matrix Multiplication
Category:Elementary Row Operations
Elementary Symmetric Function/Examples/Monic Polynomial

== Example of Elementary Symmetric Function: Monic Polynomial ==
Let $\set {x_1, x_2, \ldots, x_n}$ be a set of real or complex values, not required to be unique.
The expansion of the monic polynomial in variable $x$ with roots $\set {x_1, x_2, \ldots, x_n}$ has coefficients which are parity operators times an elementary symmetric function:
:$\ds \prod_{j \mathop = 1}^n \paren {x - x_j} = x^n - \map {e_1} {\set {x_1, \ldots, x_n} } x^{n - 1} + \map {e_2} {\set {x_1, \ldots, x_n} } x^{n - 2} - \dotsb + \paren {-1}^n \map {e_n} {\set {x_1, \ldots, x_n} }$
== Proof ==
This is an exposition of Viète's Formulas.
== Also see ==
* Viète's Formulas
Category:Elementary Symmetric Functions
Elementary Symmetric Function/Examples/m = 0

== Example of Elementary Symmetric Function: $m = 0$ ==
Let $\map {e_0} {\set {x_1, x_2, \ldots, x_n} }$ be an elementary symmetric function in $n$ variables of degree $0$.
Then:
:$\map {e_0} {\set {x_1, x_2, \ldots, x_n} } = 1$
== Proof ==
By definition:
Whether the summation $\ds \sum_{1 \mathop \le n}$ makes sense, as such, is a moot point.
Category:Elementary Symmetric Functions
Elementary Symmetric Function/Examples/m Greater than n

== Example of Elementary Symmetric Function: $m > n$ ==
Let $\map {e_m} {\set {x_1, x_2, \ldots, x_n} }$ be an elementary symmetric function in $m$ variables of degree $n$.
Let $m > n$.
Then:
== Proof ==
By definition:
However, there does not exist a set of $m$ integers which is a subset of $\set {1, 2, \ldots, n}$.
That is, the summation $\ds \sum_{1 \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_m \mathop \le n}$ is vacuous.
Hence the result.
Category:Elementary Symmetric Functions
Elements Well Inside Form Ideal

== Theorem ==
Let $L = \struct{S, \vee, \wedge, \preceq}$ be a distributive lattice with greatest element $\top$ and smallest element $\bot$.
Let $\eqslantless$ denote the well inside relation on $L$.
Then:
:$\forall a \in S : \set{b \in S: b \eqslantless a}$ is a a lattice ideal
== Proof ==
Let $a \in S$.
Let $I = \set{b \in S: b \eqslantless a}$.
=== $I$ is an Lower Section ===
Let $b \in I$ and $c \in S : c \preceq b$.
By :
:$a \preceq a$
Hence we have:
:$c \preceq b \eqslantless a \preceq a$
From Well Inside Relation Extends to Predecessor and Successor:
:$c \eqslantless a$
Hence:
:$c \in I$
It follows that $I$ is a lower section by definition.
=== $I$ is a Join Subsemilattice ===
Let $b, c \in I$.
By definition of well inside relation:
:$\exists x, y \in S : b \wedge x = \bot, a \vee x = \top, c \wedge y = \bot, a \vee y = \top$ 
We have:
Also we have:
By definition of well inside relation:
:$b \vee c \eqslantless a$
By definition of $I$:
:$b \vee c \in I$
It follows that $I$ is an join subsemilattice by definition.
It follows that $I$ is a lattice ideal by definition.
The result follows.
== Sources ==
* : Chapter $\text {III}$: Compact Hausdorff Spaces, $\S1.1$
Category:Well Inside Relation