For a genuinely nonlinear hyperbolic system of conservation laws with added artificial viscosity,
u
t
+
f
(
u
)
x
=
ε
u
x
x
{u_t} + f{(u)_x} = \varepsilon {u_{xx}}
, we prove that traveling wave profiles for small amplitude extreme shocks (the slowest and fastest) are linearly stable to perturbations in initial data chosen from certain spaces with weighted norm; i.e., we show that the spectrum of the linearized equation lies strictly in the left-half plane, except for a simple eigenvalue at the origin (due to phase translations of the profile). The weight
e
c
x
{e^{cx}}
is used in components transverse to the profile, where, for an extreme shock, the linearized equation is dominated by unidirectional convection.