  Magnetic field measurements in the upper chromosphere and above, where the
gas-to-magnetic pressure ratio $\beta$ is lower than unity, are essential for
understanding the thermal structure and dynamical activity of the solar
atmosphere. Recent developments in the theory and numerical modeling of
polarization in spectral lines have suggested that information on the magnetic
field of the chromosphere-corona transition region could be obtained by
measuring the linear polarization of the solar disk radiation at the core of
the hydrogen Lyman-$\alpha$ line at 121.6~nm, which is produced by scattering
processes and the Hanle effect. The Chromospheric Lyman-$\alpha$
Spectropolarimeter (CLASP) sounding rocket experiment aims to measure the
intensity (Stokes $I$) and the linear polarization profiles ($Q/I$ and $U/I$)
of the hydrogen Lyman-$\alpha$ line. In this paper we clarify the information
that the Hanle effect can provide by applying a Stokes inversion technique
based on a database search. The database contains all theoretical $Q/I$ and
$U/I$ profiles calculated in a one-dimensional semi-empirical model of the
solar atmosphere for all possible values of the strength, inclination, and
azimuth of the magnetic field vector, though this atmospheric region is highly
inhomogeneous and dynamic. We focus on understanding the sensitivity of the
inversion results to the noise and spectral resolution of the synthetic
observations as well as the ambiguities and limitation inherent to the Hanle
effect when only the hydrogen Lyman-$\alpha$ is used. We conclude that
spectropolarimetric observations with CLASP can indeed be a suitable diagnostic
tool for probing the magnetism of the transition region, especially when
complemented with information on the magnetic field azimuth that can be
obtained from other instruments.
