We can formalize dimension systems by considering a vector space $D$ over $\mathbb{R}$ or $\mathbb{Q}$. A dimension system is then the cartesian product $\text{Dim}_D = \mathbb{K}\times D$, with $\mathbb{K}$ some positive number system ($\mathbb{R}^+_*$ essentially) equipped with the multiplication map: $$\begin{array}{rccc} \times:\;& \text{Dim}_D\times \text{Dim}_D& \to& \text{Dim}_D\\ & \left(r_1,d_1\right)\times(r_2,d_2)&\to &\left(r_1 r_2,d_1+d_2\right), \end{array}$$ an exponent map (defined over the proper subset of $\mathbb{K}$ if it includes negative numbers) $$\begin{array}{rccc} \hat{} :\;& \text{Dim}_D\times \mathbb{R}& \to& \text{Dim}_D\\ & \left(r,d\right)^x&\to &\left(r^x,x \,d\right), \end{array}$$ the $D$-labelled addition maps acting on subspaces of $\text{Dim}_D$ with fixed dimension $\text{Dim}_D\big|_d$ $$\begin{array}{rccc} +_d:\;& \text{Dim}_D\big|_d\times \text{Dim}_D\big|_d& \to& \text{Dim}_D\big|_d\\ & \left(r_1,d\right)+(r_2,d)&\to& \left(r_1 +r_2,d\right), \end{array}$$ and the scalar multiplication $$\begin{array}{rccc} \cdot:\;& \mathbb{K}\times \text{Dim}_D& \to& \text{Dim}_D\\ & r_1\cdot(r_2,d)&\to &\left(r_1 r_2,d\right), \end{array}$$ In practice, the vector space $D$ is finite dimensional and choosing a basis $d_1,\dots,d_N$ corresponds to choosing a set of base quantities in which all others can be expressed. As an example, the SI system uses time, length, mass, electric current, temperature, amount of matter and luminous intensity as its base quantities. Having a finite-dimensional dimension system allows us to also express all quantities in our dimension system $\text{Dim}_D$ in terms of a finite number of quantities. Let us consider a basis $d_1,\dots,d_N$ over $D$, we can choose $r_1,\dots,r_N$ positive numbers in $\mathbb{K}$ and build base units $U_i = (r_i,d_i)$. Let us now consider a quantity $Q=(r,d)$ such that $d=\sum c_i d_i$. We can express $Q$ in terms of our base units as $$\begin{align} Q = \frac{r }{\prod r_i^{c_i}} \prod U_i^{c_i}, \end{align}$$ and we can therefore encode $Q$ in the unit system $\left\{U_i\right\}$ as $\left(q,c_1,\dots,c_N\right)$ with $q= r/\prod r_i$. What is commonly thought of as a change of unit in physics corresponds to a change of the $U_i$ by a change of the unit value $r_i$. A change of unit system is a change of the $U_i$ which also involves a non-diagonal basis change in $D$. Typical unit system changes are what physicists call "setting some quantity to 1", i.e. measuring some dimensionful quantity as a multiple of some characteristic quantity of the problem at hand. One very examplary case is that of high-energy physics where one chooses the speed of light as a base unit to measure velocity, the Planck action to measure angular momenta and some relevant energy scale (GeV, MeV) to measure energies. This spans the subset of SI quantities generated by lenghts, times, and masses, and just like in SI, where one can see energy as being a mass multiplied by a length squared divided by a time squared, a distance in the HEP system can be seen as a velocity times an angular momentum divided by an energy.

A slick way to handle the problem of unit system changes is to exploit the logarithm (hence the importance of positive numbers) Indeed we have $$\begin{align} \log Q= \log q + \sum c_i \log U_i, \end{align}$$ For quantities with $q=1$, we can encode $\log Q$ as just a vector $\vec c$ and immediately see that this is a nice notation. Indeed, a unit system change without rescalings, i.e. with $U_i = \prod_j \tilde U_j^{y_{ij}}$, acts as a linear transformation on the exponent vector: denoting $\vec{\tilde c} = y\cdot \vec c$, we have $\log Q = \sum \tilde c_i \log \tilde U_i$. Things get more annoying when we start considering the scaling factors either in front of quantities or in variable changes. Let us look at the basic case of rescaling of of the base units $U_1 = r \tilde U_1$, $U_i = \tilde U_i$ for $i>1$. We then have $$\begin{align} \log Q & = \log q + \sum c_i \log U_i\\ &= \log q + \log r + \sum c_i \log \tilde U_i.\\ &= \log q r + \sum c_i \log \tilde U_i \end{align}$$ Which invites us to think of the "dimensionless log" in our expression as some kind of shift on top of our "vectorial" expression consituted by the linear combination of logs of dimensionful quantities $U_i$. Furthermore, rescalings can be seen as an extra shift and this makes the analogy to an affine space attractive and that hints us at a nice way to encode dimensionful quantities: the vector $(\log \kappa,c_1,\dots,c_n)$. We then have $$\begin{align} \log Q = \left(\log q, c_1,\dots,c_N\right)\cdot \begin{pmatrix}1\\\log U_1\\\vdots\\\log U_N\end{pmatrix}. \end{align}$$ A general change of coordinate system is defined by $U_i = \kappa_i \prod \tilde U_j^{y_{ij}}$, which we can encode again as a linear transformation: $$\begin{align} \begin{pmatrix}1\\\log U_1\\\vdots\\\log U_N\end{pmatrix} &= \left(\begin{array}{c|ccc} 1&0&\dots&0\\\hline \log \kappa_1 \\ \vdots & & Y &\\ \log \kappa_N \end{array}\right)\cdot \begin{pmatrix}1\\\log \tilde U_1\\\vdots\\\log \tilde U_N\end{pmatrix}. \end{align}$$ We can then define the action of the change of system on the vector representation of the quantity as $$\begin{align} \left(\log \tilde q, \tilde c_1,\dots,\tilde c_N\right)= \left(\log q, c_1,\dots,c_N\right)\cdot \left(\begin{array}{c|ccc} 1&0&\dots&0\\\hline \log \kappa_1 \\ \vdots & & Y &\\ \log \kappa_N \end{array}\right), \end{align}$$ yielding $$\begin{align} \log Q = \left(\log \tilde q, \tilde c_1,\dots,\tilde c_N\right) \cdot \begin{pmatrix}1\\\log \tilde U_1\\\vdots\\\log \tilde U_N\end{pmatrix}. \end{align}$$ This matrix notation is of course very convenient as it encodes all possible transformation in a unified language, as well as reduces the chaining of changes of systems to a chain of matrix multiplications.