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In the following definition, we introduce the notion of a “matrix” from a pure notational perspective, without in any way considering interesting mathematical properties of matrices, like dimensions and basis, which we will deal later in detail.

Definition: Matrix, Set of Matrices over a Field

Let \(F\) be a field and let \(\alpha_{ij}\in F\) be arbitrary field elements for \(i=1,\ldots,m\), \(j=1,\ldots,n\). Then the structure

\[
A:=\pmatrix{
\alpha_{11} & \alpha_{12} & \ldots & \alpha_{1n} \cr
\alpha_{21} & \alpha_{22} & \ldots & \alpha_{2n} \cr
\vdots & \vdots & \ddots & \vdots \cr
\alpha_{m1} & \alpha_{m2} & \ldots & \alpha_{mn} \cr
}
\]

is called a matrix over the field \(F\) with \(m\) rows and \(n\) columns.

The set of all matrices over the field \(F\) with \(m\) rows and \(n\) columns is denoted by \(M_{m\times n}(F)\).

Generalization

If \(I\) and \(J\) are index sets, then the matrix \(I\times J\) is a function of the form

\(I\times J\longrightarrow F,\,(i,j)\longmapsto a_{ij}\,.\)

and can be visualized as
\[{\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}&\ldots\\a_{21}&a_{22}&\ldots &a_{2n}&\ldots\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}&\ldots\\
\vdots&\vdots&\ldots &\vdots&\ddots\end{pmatrix}}\]

| | | | | created: 2014-11-02 20:42:15 | modified: 2018-04-09 23:41:12 | by: bookofproofs | references: [979], [6907]

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[6907] Brenner, Prof. Dr. rer. nat., Holger: “Various courses at the University of Osnabrück”, https://de.wikiversity.org/wiki/Wikiversity:Hochschulprogramm, 2014

Bibliography (further reading)

[979] Reinhardt F., Soeder H.: “dtv-Atlas zur Mathematik”, Deutsche Taschenbuch Verlag, 1994, 10