The vector space $M_{m\times n}(F):=F^{m\times n}$ of $m$ by $n$ matrices over a field, with entrywise operations

Let $F$ be a field (Field) and let $m,n\in \mathbb{N}$. Recall that a natural number is a von Neumann natural (The natural numbers $\mathbb{N}$ (von Neumann)), so $m=\{0,\dots ,m-1\}$ and $n=\{0,\dots ,n-1\}$ (On $\mathbb{N}$ the order is membership: $m<n⟺m\in n$), and let $m\times n$ be their cartesian product. An $m$ by $n$ matrix over $F$ is an element of the function space $F^{m\times n}$ (The vector space $F^X$ of all functions $X\to F$ with pointwise operations, and $F^n$ as the case $X=n=\{0,1,\dots ,n-1\}$), that is a function $a:m\times n\to F$; we write

$a_{ij}:=a(i,j)(i<m,j<n)$

for its entries, $i$ being the row index and $j$ the column index. Write

$M_{m\times n}(F):=F^{m\times n},$

and $M_n(F):=M_{n\times n}(F)$ for the square case.

Since $M_{m\times n}(F)$ is the function space $F^{m\times n}$, it is a vector space over $F$ with the pointwise operations of The vector space $F^X$ of all functions $X\to F$ with pointwise operations, and $F^n$ as the case $X=n=\{0,1,\dots ,n-1\}$, which read entrywise:

$(a+b{)}_{ij}=a_{ij}+b_{ij},(\lambda a{)}_{ij}=\lambda a_{ij}(i<m,j<n),$

and the zero of $M_{m\times n}(F)$ is the matrix all of whose entries are $0_F$. No verification is needed beyond that already carried out in The vector space $F^X$ of all functions $X\to F$ with pointwise operations, and $F^n$ as the case $X=n=\{0,1,\dots ,n-1\}$ for an arbitrary index set: this is the case $X=m\times n$.

Both indices start at $0$. The rows are indexed by $i\in m=\{0,\dots ,m-1\}$ and the columns by $j\in n=\{0,\dots ,n-1\}$, so the entries of a $2$ by $3$ matrix are $a_{00},a_{01},a_{02},a_{10},a_{11},a_{12}$.

The degenerate shapes. $\mathbb{N}$ contains $0$, so $m=0$ and $n=0$ are genuine cases. If $m=0$ or $n=0$ then $m\times n=\mathrm{\varnothing }$, so $M_{m\times n}(F)$ has exactly one element, the empty function, and is the zero space; there is no matrix of shape $0$ by $n$ other than that one.

• This item introduces the vector-space structure and the notation, and nothing else. There is no matrix product here, no identity matrix, no ring $M_n(F)$, no determinant and no matrix of a linear map. Those belong to a later page, which must add the product to this object rather than introduce a second notion of matrix; the addition and the scalar multiplication used there are the entrywise ones defined above.

• A matrix is a function, so equality of matrices is equality of functions: $a=b$ exactly when $a_{ij}=b_{ij}$ for all $i<m$ and $j<n$. Nothing below relies on a matrix being written as a rectangular array; the array is a way of displaying the function.

• Why the shape is a pair of naturals rather than a pair of finite index sets. Taking the index set to be $m\times n$ with $m$ and $n$ natural numbers keeps the coordinates of a matrix, of a tuple in $F^n$ (The vector space $F^X$ of all functions $X\to F$ with pointwise operations, and $F^n$ as the case $X=n=\{0,1,\dots ,n-1\}$) and the finite sums of The product $g_0{g}_1\cdots g_{n-1}$ of a finite list in a monoid, by recursion, with the empty product ($n=0$) equal to the identity all indexed by the same kind of object, which is what makes the boundary cases above statable at all.