A matrix is a way of storing numbers with two indices, but the real mathematical question is what kind of object the whole collection of such arrays forms. If we only look at one matrix at a time, we can solve a linear system or compute a determinant. If we look at all matrices of a fixed size at once, we get a [vector space](/page/Vector%20Space) whose coordinates are entries, whose points can represent linear maps, and whose subspaces encode meaningful constraints.

The first surprise is that a matrix has several lives. A single $m \times n$ array may be a coefficient table, a vector in an $mn$-dimensional space, or the coordinate representative of a [linear map](/page/Linear%20Map) from $k^n$ to $k^m$. Confusing these roles leads to mistakes: rectangular matrices form vector spaces even when they cannot be multiplied among themselves, while square matrices have an additional multiplication that is usually noncommutative.

[example: A Linear System as a Point] Let $k$ be a field, and consider the coefficient matrix $A\in M_{2\times 3}(k)$ with entries $A_{11}=1$, $A_{12}=-2$, $A_{13}=0$, $A_{21}=3$, $A_{22}=1$, and $A_{23}=4$. The corresponding system in unknowns $x_1,x_2,x_3$ has left-hand sides \begin{align*} A_{11}x_1+A_{12}x_2+A_{13}x_3=x_1-2x_2+0x_3 \end{align*} and \begin{align*} A_{21}x_1+A_{22}x_2+A_{23}x_3=3x_1+x_2+4x_3. \end{align*} As a point of the matrix space $M_{2\times 3}(k)$, the same matrix is assembled entry by entry from the matrix units: \begin{align*} A=A_{11}E_{11}+A_{12}E_{12}+A_{13}E_{13}+A_{21}E_{21}+A_{22}E_{22}+A_{23}E_{23}. \end{align*} Substituting the six entries gives \begin{align*} A=1E_{11}+(-2)E_{12}+0E_{13}+3E_{21}+1E_{22}+4E_{23}. \end{align*} Removing the zero term and the coefficient $1$ terms gives \begin{align*} A=E_{11}-2E_{12}+3E_{21}+E_{22}+4E_{23}. \end{align*} It also defines a linear map $T_A:k^3\to k^2$ by $T_A(x)=Ax$. For $x=(x_1,x_2,x_3)\in k^3$, matrix multiplication gives \begin{align*} T_A(x)=(x_1-2x_2,\;3x_1+x_2+4x_3). \end{align*} Thus the same object is a coefficient table, a coordinate vector in the six-dimensional space $M_{2\times 3}(k)$, and the coordinate representative of a map from $k^3$ to $k^2$. [/example]

The purpose of matrix space is to keep those roles organized. We first build the vector space of all matrices of a fixed size, then connect it to linear maps, then examine important subspaces and nonlinear rank conditions. The chapter ends by explaining what extra structure appears over $\mathbb{R}$ and $\mathbb{C}$.

## Definition

The basic construction answers a simple coordinate question: if each entry of an $m$ by $n$ array may vary independently in a field $k$, what vector space do those arrays form? We need this ambient space before discussing special matrices, matrix equations, or matrices as representatives of maps.

[definition: Matrix Space] Let $k$ be a field, and let $m,n \in \mathbb{N}$. The matrix space $M_{m \times n}(k)$ is the set of all $m \times n$ matrices $A=(A_{ij})$ with entries $A_{ij}\in k$ for $1\le i\le m$ and $1\le j\le n$. Addition and scalar multiplication are defined entrywise by $(A+B)_{ij}=A_{ij}+B_{ij}$ and $(\lambda A)_{ij}=\lambda A_{ij}$. [/definition]

This definition deliberately does not mention matrix multiplication. Rectangular matrices still form a vector space, even though multiplying two arbitrary elements of $M_{m\times n}(k)$ is not usually defined. What is available is compatible multiplication across shapes: a matrix in $M_{m\times n}(k)$ can multiply a matrix in $M_{n\times p}(k)$ to produce an element of $M_{m\times p}(k)$. Later sections use the shorter notation $M_n(k)$ for the square case, where multiplication, powers, inverses, traces, and determinants all stay inside one fixed space.

## Coordinate Structure

To compute in matrix space, we need coordinate axes. The natural axes are the matrices that switch on exactly one entry and set all other entries to zero, because every matrix can then be assembled entry by entry.

[definition: Matrix Unit] Let $k$ be a field and let $m,n\in\mathbb{N}$. For $1\le i\le m$ and $1\le j\le n$, the matrix unit $E_{ij}\in M_{m\times n}(k)$ is the matrix whose $(i,j)$-entry is $1$ and whose other entries are $0$. [/definition]

Matrix units are the standard coordinate vectors of matrix space. They allow us to write every matrix as a finite coordinate expansion:

\begin{align*} A=\sum_{i=1}^{m}\sum_{j=1}^{n}A_{ij}E_{ij}. \end{align*}

This is the same idea as writing a vector in $k^d$ as a linear combination of standard basis vectors, but now the single coordinate index has been replaced by a pair of indices.

The expansion above would be misleading if there were hidden linear relations among the matrix units, or if some matrices could not be reached by entry-by-entry assembly. What must be checked is that choosing the $mn$ entries freely gives exactly $mn$ independent directions in $M_{m\times n}(k)$.

This turns the coordinate formula into a dimension statement: the matrix units should not merely describe entries, but should provide a genuine basis for the whole matrix space. The spanning part says that entries really do assemble every matrix, while the independence part says that no entry position can be forced by the others. Proving both points is what turns the visual grid of coordinate positions into a reliable coordinate system for $M_{m\times n}(k)$.

The next formal result is needed to promote the entry-by-entry description from useful notation to an actual coordinate theorem. It answers the structural question raised by the definition: whether the listed matrices $E_{ij}$ account for the entire vector space, with no redundancy among the coordinate positions.

[quotetheorem:8369]

The theorem is the main reason matrix spaces behave like familiar finite-dimensional coordinate spaces. It also tells us exactly how many independent parameters an $m\times n$ matrix has.

## Coordinates and Entry Functionals