  This paper provides versions of classical results from linear algebra, real
analysis and convex analysis in a free module of finite rank over the ring
$L^0$ of measurable functions on a $\sigma$-finite measure space. We study the
question whether a submodule is finitely generated and introduce the more
general concepts of $L^0$-affine sets, $L^0$-convex sets, $L^0$-convex cones,
$L^0$-hyperplanes, $L^0$-half-spaces and $L^0$-convex polyhedral sets. We
investigate orthogonal complements, orthogonal decompositions and the existence
of orthonormal bases. We also study $L^0$-linear, $L^0$-affine, $L^0$-convex
and $L^0$-sublinear functions and introduce notions of continuity,
differentiability, directional derivatives and subgradients. We use a
conditional version of the Bolzano-Weierstrass theorem to show that conditional
Cauchy sequences converge and give conditions under which conditional
optimization problems have optimal solutions. We prove results on the
separation of $L^0$-convex sets by $L^0$-hyperplanes and study $L^0$-convex
conjugate functions. We provide a result on the existence of $L^0$-subgradients
of $L^0$-convex functions, prove a conditional version of the Fenchel-Moreau
theorem and study conditional inf-convolutions.
