  An emerging theory of "linear-algebraic pseudorandomness" aims to understand
the linear-algebraic analogs of fundamental Boolean pseudorandom objects where
the rank of subspaces plays the role of the size of subsets. In this work, we
study and highlight the interrelationships between several such algebraic
objects such as subspace designs, dimension expanders, seeded rank condensers,
two-source rank condensers, and rank-metric codes. In particular, with the
recent construction of near-optimal subspace designs by Guruswami and Kopparty
as a starting point, we construct good (seeded) rank condensers (both lossless
and lossy versions), which are a small collection of linear maps $\mathbb{F}^n
\to \mathbb{F}^t$ for $t \ll n$ such that for every subset of $\mathbb{F}^n$ of
small rank, its rank is preserved (up to a constant factor in the lossy case)
by at least one of the maps.
  We then compose a tensoring operation with our lossy rank condenser to
construct constant-degree dimension expanders over polynomially large fields.
That is, we give $O(1)$ explicit linear maps $A_i:\mathbb{F}^n\to \mathbb{F}^n$
such that for any subspace $V \subseteq \mathbb{F}^n$ of dimension at most
$n/2$, $\dim\bigl( \sum_i A_i(V)\bigr) \ge (1+\Omega(1)) \dim(V)$. Previous
constructions of such constant-degree dimension expanders were based on
Kazhdan's property $T$ (for the case when $\mathbb{F}$ has characteristic zero)
or monotone expanders (for every field $\mathbb{F}$); in either case the
construction was harder than that of usual vertex expanders. Our construction,
on the other hand, is simpler.
  Via an equivalence to linear rank-metric codes, we then construct optimal
lossless two-source condensers. We then use our seeded rank condensers to
obtain near-optimal lossy two-source condensers for constant rank sources.
