  Recently, the worse-case analysis, probabilistic analysis and empirical
justification have been employed to address the fundamental question: When does
$\ell_1$-minimization find the sparsest solution to an underdetermined linear
system? In this paper, a deterministic analysis, rooted in the classic linear
programming theory, is carried out to further address this question. We first
identify a necessary and sufficient condition for the uniqueness of least
$\ell_1$-norm solutions to linear systems. From this condition, we deduce that
a sparsest solution coincides with the unique least $\ell_1$-norm solution to a
linear system if and only if the so-called \emph{range space property} (RSP)
holds at this solution. This yields a broad understanding of the relationship
between $\ell_0$- and $\ell_1$-minimization problems. Our analysis indicates
that the RSP truly lies at the heart of the relationship between these two
problems. Through RSP-based analysis, several important questions in this field
can be largely addressed. For instance, how to efficiently interpret the gap
between the current theory and the actual numerical performance of
$\ell_1$-minimization by a deterministic analysis, and if a linear system has
multiple sparsest solutions, when does $\ell_1$-minimization guarantee to find
one of them? Moreover, new matrix properties (such as the \emph{RSP of order
$K$} and the \emph{Weak-RSP of order $K$}) are introduced in this paper, and a
new theory for sparse signal recovery based on the RSP of order $K$ is
established.
