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 ${\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}&\ldots\\a_{21}&a_{22}&\ldots &a_{2n}&\ldots\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}&\ldots\\ \vdots&\vdots&\ldots &\vdots&\ddots\end{pmatrix}}$

Chapters: 1 2
Definitions: 3 4 5 6 7 8 9
Examples: 10 11
Explanations: 12
Proofs: 13 14
Propositions: 15

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@Brenner

### References

#### Bibliography

1. Reinhardt F., Soeder H.: "dtv-Atlas zur Mathematik", Deutsche Taschenbuch Verlag, 1994, 10th Edition

#### Adapted from CC BY-SA 3.0 Sources:

1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück