W^3
Posts:
29
Registered:
4/19/11


Re: Matrices of rank at least k
Posted:
Nov 29, 2012 1:15 PM


Let S = {a in R^k : a = 1}. Suppose u_1, ..., u_k are linearly independent in R^m. Then for all a in S, sum a_ju_j> 0 (the sum will always be over j in {1, ..., k}). By continuity and compactness, it follows that
inf_{a in S} sum a_ju_j = c > 0.
Let r = c/2k. Suppose v_j is in B(u_j,r), j = 1, ..., k (these are open balls in R^m). Then
sum a_jv_j >= sum a_ju_j  sum a_j(v_ju_j)
>= c  kr = c/2 > 0
for all a in S. This shows v_1, ..., v_k are linearly independent in R^m, which gives the desired result.

