  In this paper, we investigate the degrees of freedom ($\dof$) of penalized
$\ell_1$ minimization (also known as the Lasso) for linear regression models.
We give a closed-form expression of the $\dof$ of the Lasso response. Namely,
we show that for any given Lasso regularization parameter $\lambda$ and any
observed data $y$ belonging to a set of full (Lebesgue) measure, the
cardinality of the support of a particular solution of the Lasso problem is an
unbiased estimator of the degrees of freedom. This is achieved without the need
of uniqueness of the Lasso solution. Thus, our result holds true for both the
underdetermined and the overdetermined case, where the latter was originally
studied in \cite{zou}. We also show, by providing a simple counterexample, that
although the $\dof$ theorem of \cite{zou} is correct, their proof contains a
flaw since their divergence formula holds on a different set of a full measure
than the one that they claim. An effective estimator of the number of degrees
of freedom may have several applications including an objectively guided choice
of the regularization parameter in the Lasso through the $\sure$ framework. Our
theoretical findings are illustrated through several numerical simulations.
