. We have to prove the existence of a subset
of cardinality at most
, choose a linear map
(some projection on a complement subspace, for instance). Finally, let us introduce the linear map
. Its kernel is
due to the assumption, hence
is injective, and its image has dimension
. Consider a matrix of
, consisting in piling up the matrices of
); then we can find
independent rows. Let
be the subspaces corresponding to these rows (
if two rows come from the same "submatrix"). The map
by the previous choice, hence its kernel (i.e.
. This concludes.