  We consider the problem of recovering a partially sparse solution of an
underdetermined system of linear equations by minimizing the $\ell_1$-norm of
the part of the solution vector which is known to be sparse. Such a problem is
closely related to a classical problem in Compressed Sensing where the
$\ell_1$-norm of the whole solution vector is minimized. We introduce analogues
of restricted isometry and null space properties for the recovery of partially
sparse vectors and show that these new properties are implied by their original
counterparts. We show also how to extend recovery under noisy measurements to
the partially sparse case.
