  Every (left) linear function on a subspace of a finite-dimensional vector
space over a (skew) field can be extended to a (left) linear function on the
whole space. This paper explores the extent to what this basic fact of linear
algebra is applicable to more general structures. Semifields with a similar
property imposed on linear functions are called (left) exact, and we present a
complete description of such semifields. Namely, we show that a semifield $S$
is left exact if and only if $S$ is either a skew field or an idempotent
semiring. In particular, our result is new even for the tropical semiring and
gives a solution to the problem posed by Wilding. Also, we point out several
problems that require further investigation.
