The aim of this paper is to develop the exact technique to solve the time optimization problem for the linear control system \[ \dot x(t)= Ax(t)+ bu(t)+ f(t),\quad x\in\mathbb{R}^n,\quad x(0)= x_0,\quad x(T)= x_1, \] with \(n\times n\) matrix \(A\) and \(n\)-vector \(b\), both constant, and a given vector function \(f(t)\). The author improves the conventional approach to solving such problems, which is related to well-known moment problem, and establishes a relationship of the time optimization problem with the basic linear approximation problem in the space \(L_p\), \(1\leq p<\infty\). On this basis, he gives the algebraic solution to the above problem for the case of Hilbert spaces. In conclusion, the exact solution of the time optimization problem is obtained for the second-order homogeneous control system.