I have a sequence of $m$-dimensional vector spaces indexed by $t$ $$S_t = \text{span}\left(\vec{v}_{1,t},\vec{v}_{2,t},\ldots,\vec{v}_{m,t}\right)$$ All the $S_t$ are subspaces of an $n$-dimensional vector space with $n\gg m$.
I have reason to believe that these spaces do not change much with $t$, and I would like to make a linear approximation $$S_t \approx \text{span}\left(\vec{a}_1+\vec{b}_1 t,\vec{a}_2+\vec{b}_2 t,\ldots,\vec{a}_m+\vec{b}_m t\right)$$ i.e. I would like to find the $\vec{a}_i, \vec{b}_i$ such that the above expression becomes the "best fit" in some sense.
Is there some natural way of solving this problem?
If the $S_t$ could be coordinatized in some way, i.e. if the space of $m$-dimensional subspaces of the overall $n$-dimensional vector space could itself be given the structure of a normed vector space, that would solve the problem, but it is not clear to me that that can be done.
Another possibility is to construct some sort of distance metric. Let $P_t$ be the projection operator on $S_t$ ($P_t$ is a matrix), then we could define a distance function $$d(S_i,S_j)=\|P_i-P_j\|$$ Conceivably $\vec{a}_i, \vec{b}_i$ could be obtained by minimizing the (square of) the total thus-defined "distance" between the given $S_t$ and the linearly approximated $S_t$, but that seems like an overly complicated approach.