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".
When \(R\) is any field, with the scalar multiplication defined as \((r M)^j_l = r M^j_l\), the set of the \(m \times n\) matrices is a vectors space over \(R\) 'vectors spaces - linear morphisms' isomorphic to \(R^{m n}\).
