  We address the problem of recovering a sparse $n$-vector within a given
subspace. This problem is a subtask of some approaches to dictionary learning
and sparse principal component analysis. Hence, if we can prove scaling laws
for recovery of sparse vectors, it will be easier to derive and prove recovery
results in these applications. In this paper, we present a scaling law for
recovering the sparse vector from a subspace that is spanned by the sparse
vector and $k$ random vectors. We prove that the sparse vector will be the
output to one of $n$ linear programs with high probability if its support size
$s$ satisfies $s \lesssim n/\sqrt{k \log n}$. The scaling law still holds when
the desired vector is approximately sparse. To get a single estimate for the
sparse vector from the $n$ linear programs, we must select which output is the
sparsest. This selection process can be based on any proxy for sparsity, and
the specific proxy has the potential to improve or worsen the scaling law. If
sparsity is interpreted in an $\ell_1/\ell_\infty$ sense, then the scaling law
can not be better than $s \lesssim n/\sqrt{k}$. Computer simulations show that
selecting the sparsest output in the $\ell_1/\ell_2$ or thresholded-$\ell_0$
senses can lead to a larger parameter range for successful recovery than that
given by the $\ell_1/\ell_\infty$ sense.
