A matrix is usually informally defined as a rectangular array of numbers. To make this definition formal, we can define a matrix as a map from $\{1,…,m\} \times \{1,…,n\}$ to the underlying field of scalars, where $\times$ denotes cartesian product. However, a subtle complication arises when $m=0$ or $n=0$. In that case, the matrix would be an empty function. The problem, however, is that there is then no way to distinguish between $m \times 0$ matrices from $0 \times n$ matrices. In fact, under the cartesian product definition, for all natural numbers $m$, $m'$, $n$, and $n'$, the $m \times 0$, $m' \times 0$, $0 \times n$, and $0 \times n'$ matrices are all the same entity, namely the empty function. This is, to me, an undesirable state of affairs. I want to be able to distinguish, for example, $2 \times 0$, $3 \times 0$, $0 \times 2$, and $0 \times 3$ matrices. Is there a better definition of matrix that some mathematician has written about in some paper or book that avoids that problem?

# Formal definition of $n$ by $0$ and $0$ by $n$ matrices

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## Best Answer

A $m\times n$ matrix is a representation of a mapping from a $n$-dimensional vector space to a $m$-dimensional vector space. In that sense, a $0\times m$ matrix

isdifferent from a $n\times 0$ matrix. While they both represent the mapping we call "the zero mapping", the zero mappings aredifferentmappings.In other words, instead of speaking of mappings from $\{1,\dots,m\}\times\{1,\dots,n\}$, you can speak of

linear mapsfrom $\mathbb F^n$ to $\mathbb F^m$ (usually denoted something like $\mathcal L(\mathbb F^n, \mathbb F^m)$), and instead of speaking of a $0\times m$ matrix, you can speak of the element of $\mathcal L(\mathbb F^0, \mathbb F^n)$. That element (there is only one) is different from the element of $\mathcal L(\mathbb F^m, \mathbb F^0)$.