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.
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 \(M_{m\times n}(F)\).
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
