  We consider the method of alternating (metric) projections for pairs of
linear subspaces of finite dimensional Banach spaces. We investigate the size
of the set of points for which this method converges to the metric projection
onto the intersection of these subspaces. In addition we give a
characterisation of the pairs of subspaces for which the alternating projection
method converges to the projection onto the intersection for every initial
point. We provide a characterisation of the linear subspaces of $\ell_p^n$,
$1<p<\infty$, $p\neq 2$, which admit a linear metric projection and use this
characterisation to show that in $\ell_p^3$, $1<p<\infty$, $p\neq 2$, the set
of pairs of subspaces for which the alternating projection method converges to
the projection onto the intersection is small in a probabilistic sense.
