Basic linear algebra (Gauss-Jordan etc.)
takes a set of -equations-,
and -parametrizes- the space of solutions.

Dually, given a vector space,
you can find defining equations
(and indeed -parametrize- the space of defining equations)
by dualizing!

Concretely, by taking the transpose and doing Gauss-Jordan.

More precisely:
a set of equations defines a subspace;
you can find a -basis- for it
(and then all bases are a $\GL_k$ torsor over this)

dually, given a subspace,
the equations that vanish on it are its annihilator,
which is a subspace of the dual,
and you can proceed identically.

--------------------------------------
affineness (equations that don't equal zero)
makes this slightly trickier, but not much
(just tack on another variable).
