  Let $k \subset K$ be an extension of fields, and let $A \subset M_{n}(K)$ be
a $k$-algebra. We study parameter spaces of $m$-dimensional subspaces of
$K^{n}$ which are invariant under $A$. The space $\mathbb{F}_{A}(m,n)$, whose
$R$-rational points are $A$-invariant, free rank $m$ summands of $R^{n}$, is
well known. We construct a distinct parameter space, $\mathbb{G}_{A}(m,n)$,
which is a fiber product of a Grassmannian and the projectivization of a vector
space. We then study the intersection $\mathbb{F}_{A}(m,n) \cap
\mathbb{G}_{A}(m,n)$, which we denote by $\mathbb{H}_{A}(m,n)$. Under suitable
hypotheses on $A$, we construct affine open subschemes of $\mathbb{F}_{A}(m,n)$
and $\mathbb{H}_{A}(m,n)$ which cover their $K$-rational points. We conclude by
using $\mathbb{F}_{A}(m,n)$, $\mathbb{G}_{A}(m,n)$, and $\mathbb{H}_{A}(m,n)$
to construct parameter spaces of two-sided subspaces of two-sided vector
spaces.
