See these papers:
Covering by Complements of Subspaces, II., W. Edwin Clark and Boris Shekhtman, Proc. Amer. Math. Soc.125 (1997), no. 1, 251--254.
Covering by Complements of Subspaces, W. Edwin Clark and Boris Shekhtman, Linear and Multilin. Algebra, Vol 49,1995, pp. 1--13.