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Math Forum » Discussions » sci.math.* » sci.math.independent

Topic: Matrices of rank at least k
Replies: 12   Last Post: Nov 29, 2012 1:15 PM

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W^3

Posts: 28
Registered: 4/19/11
Re: Matrices of rank at least k
Posted: Nov 29, 2012 1:15 PM
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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_j-u_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.

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