  We study the model theory of vector spaces with a bilinear form over a fixed
field. For finite fields this can be, and has been, done in the classical
framework of full first-order logic. For infinite fields we need different
logical frameworks. First we take a category-theoretic approach, which requires
very little set-up. We show that linear independence forms a simple unstable
independence relation. With some more work we then show that we can also work
in the framework of positive logic, which is much more powerful than the
category-theoretic approach and much closer to the classical framework of full
first-order logic. We fully characterise the existentially closed models of the
arising positive theory. Using the independence relation from before we
conclude that the theory is simple unstable, in the sense that dividing has
local character but there are many distinct types. We also provide positive
version of what is commonly known as the Ryll-Nardzewski theorem for
$\omega$-categorical theories in full first-order logic, from which we conclude
that bilinear spaces over a countable field are $\omega$-categorical.
