definition of %ring name% matrices space
Topics
About: matrices space
The table of contents of this article
Starting Context
- The reader knows a definition of ring.
Target Context
- The reader will have a definition of %ring name% matrices space.
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.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( R\): \(\in \{\text{ the rings }\}\)
\(*\{M\}\): \(M = \begin{pmatrix} M^j_l \end{pmatrix}\) where \(j \in \{1, ..., m\}\) and \(l \in \{1, ..., n\}\) and \(M^j_l \in R\), called "\(m \times n\) matrix", with the addition and the multiplication specified below
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Conditions:
\(\forall M = \begin{pmatrix} M^j_l \end{pmatrix}, M' = \begin{pmatrix} M'^j_l \end{pmatrix} \in \{\text{ the } m \times n \text{ matrices } \} (M + M' = \begin{pmatrix} M^j_l + M'^j_l \end{pmatrix})\)
\(\land\)
\(\forall M = \begin{pmatrix} M^j_l \end{pmatrix} \in \{\text{ the } m \times n \text{ matrices }\}, \forall M' = \begin{pmatrix} M'^l_m \end{pmatrix} \in \{\text{ the } n \times o \text{ matrices }\} (M M' = \begin{pmatrix} M^j_l M'^l_m \end{pmatrix} \in \{\text{ the } m \times o \text{ matrices }\})\)
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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".
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".
