The collection of marked unimodular lattices in the euclidean $n$-space corresponds to the symmetric space $S_n:=SO_n(\mathbb{R}) \backslash SL_n(\mathbb{R})$.

I have only recently learned that this symmetric space has the further property of being $linear$. That is to say, (i) there exists a homeomorphism $h$ of $S_n$ onto the interior of some euclidean $N$-ball $D^N$ and (ii) there is a map $i: SL_n(\mathbb{R}) \to GL_{N+1}(\mathbb{R})$ that makes the homeomorphism $h$ equivariant.

Now I do not know what the ''linearity'' of $S_n$ $really$ means. I would like to know how it relates to the following very remarkable property of unimodular lattices:

Take an unmarked rank $n$ unimodular lattice $\Gamma$. (ie. $\Gamma$ is an element in $S_n / SL_n(\mathbb{Z})$). The remarkable fact is this: for a given rational subspace $W$ in $\Gamma \otimes \mathbb{R}$ there is a CANONICAL complementary rational subspace $W^o$ --( of course, $W^o$ is just the orthogonal complement of $W$ in $\Gamma \otimes \mathbb{R}$).

My question is then how do we see the LINEARITY of $S_n$ as responsible for the fact that rational subspaces have CANONICAL rational complements? Any remarks on what a ''linear'' symmetric space really $is$ would be appreciated.