Imagine we have a vector ${\bf d} \in \mathbb N^n$. Now imagine we have another vector $${\bf w} = [\log(2),\log(3),\log(5),\log(7),\cdots,\log(p_n)]^T$$
$\exp({\bf w^Td})$ will then be the integer with it's prime factor decomposition (exponents) in $\bf d$.
- ${\bf d}_1$ will be the exponent for $2$
- ${\bf d}_2$ will be the exponent for $3$
and so on...
One interesting goal would be to be able to express number theoretic divisibility propositions with only linear algebra. There probably exist many other ways to do this, but I lack the knowledge of them. Also multiplication becomes addition of exponents so in some sense we have made multiplication linear.
But how to get rid of the pesky non-linear "exp"?