# The Unapologetic Mathematician

## The Category of Matrices III

At long last, let’s get back to linear algebra. We’d laid out the category of matrices $\mathbf{Mat}(\mathbb{F})$, and we showed that it’s a monoidal category with duals.

Now here’s the really important thing: There’s a functor $\mathbf{Mat}(\mathbb{F})\rightarrow\mathbf{FinVec}(\mathbb{F})$ that assigns the finite-dimensional vector space $\mathbb{F}^n$ of $n$-tuples of elements of $\mathbb{F}$ to each object $n$ of $\mathbf{Mat}(\mathbb{F})$. Such a vector space of $n$-tuples comes with the basis $\left\{e_i\right\}$, where the vector $e_i$ has a ${1}$ in the $i$th place and a ${0}$ elsewhere. In matrix notation:

$\displaystyle e_1=\begin{pmatrix}1\\{0}\\\vdots\\{0}\end{pmatrix}$
$\displaystyle e_2=\begin{pmatrix}{0}\\1\\\vdots\\{0}\end{pmatrix}$

and so on. We can write $e_i=\delta_i^je_j$ (remember the summation convention), so the vector components of the basis vectors are given by the Kronecker delta. We will think of other vectors as column vectors.

Given a matrix $\left(t_i^j\right)\in\hom(m,n)$ we clearly see a linear transformation from $\mathbb{F}^m$ to $\mathbb{F}^n$. Given a column vector with components $v^i$ (where the index satisfies $1\leq i\leq m$), we construct the column vector $t_i^jv^i$ (here $1\leq j\leq n$). But we’ve already established that matrix multiplication represents composition of linear transformations. Further, it’s straightforward to see that the linear transformation corresponding to a matrix $\left(\delta_i^j\right)$ is the identity on $\mathbb{F}^n$ (depending on the range of the indices on the Kronecker delta). This establishes that we really have defined a functor.

But wait, there’s more! The functor is linear over $\mathbb{F}$, so it’s a functor enriched over $\mathbb{F}$. The Kronecker product of matrices corresponds to the monoidal product of linear transformations, so the functor is monoidal, too. Following the definitions, we can even find that our functor preserves duals.

So we’ve got a functor from our category of matrices to the category of finite-dimensional vector spaces, and it preserves all of the relevant structure.