Let $G$ be a finite subgroup of $GL(4,\mathbb{Q} )$. Let
$G$ act on the rational function field
$\mathbb{Q}(x_1,x_2,x_3,x_4)$ by $\mathbb{Q}$-automorphism
defined by the linear action of variables.Linear Noether's
problem asks whether the fixed field $\mathbb{Q}
(x_1,x_2,x_3,x_4)^G$ is rational (i.e. purely transcendental)
over $\mathbb{Q}$. So far some partial results have been
known, but in this paper we will give the almost complete
results of this problem. One of motivations of this problem is
the relation to the inverse Galois problem.