1138: %Ring Name% Matrices Space

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definition of %ring name% matrices space

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Topics

About: matrices space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of ring.

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Target Context

• The reader will have a definition of %ring name% matrices space.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$R$: $\in \{\text{ the rings }\}$
$\ast \{M\}$: $M=\left(\begin{array}{c}{M}_l^j\end{array}\right)$ where $j\in \{1,...,m\}$ and $l\in \{1,...,n\}$ and $M_l^j\in R$, called "$m\times n$ matrix", with the addition and the multiplication specified below
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Conditions:
$\mathrm{\forall }M=\left(\begin{array}{c}{M}_l^j\end{array}\right),M^{\prime }=\left(\begin{array}{c}{M}_l^{\prime j}\end{array}\right)\in \{\text{ the }m\times n\text{ matrices }\}(M+M^{\prime }=\left(\begin{array}{c}{M}_l^j+M_l^{\prime j}\end{array}\right))$
$\wedge$
$\mathrm{\forall }M=\left(\begin{array}{c}{M}_l^j\end{array}\right)\in \{\text{ the }m\times n\text{ matrices }\},\mathrm{\forall }{M}^{\prime }=\left(\begin{array}{c}{M}_m^{\prime l}\end{array}\right)\in \{\text{ the }n\times o\text{ matrices }\}(M{M}^{\prime }=\left(\begin{array}{c}{M}_l^j{M}_m^{\prime l}\end{array}\right)\in \{\text{ the }m\times o\text{ matrices }\})$
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When $R$ is $\mathbb{R}$, $M$ is called "real matrix".

When $R$ is $\mathbb{C}$, $M$ is called "complex matrix".

When $m=1$, $M$ is called "row vector".

When $n=1$, $M$ is called "column vector".

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2: Note

Typically, $R$ is $\mathbb{R}$ or $\mathbb{C}$, but not necessarily so.

But usually (at least, this definition requires so) $R$ needs to be a ring, because otherwise, addition or multiplication of matrices would not be valid: if any addition or any multiplication is not needed, any $m\times n$ arrangement of any objects can be called "matrix".

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References

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