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Set of vectors with each subset forming a basis 
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#1
Nov2412, 02:29 PM

P: 78

Hey!
Let M and N be two natural numbers and N>M. I want to build a set A with N vectors of size M such that each subset S of A, where S = M, contains linearly independent vectors. Another way to put it is that every S should be a basis for R^M. Any ideas? Thanks! 


#2
Nov2612, 01:00 PM

P: 350

Do you want an explicit construction or a proof that such a set exists?



#3
Nov2712, 08:37 AM

Math
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Sci Advisor
Thanks
PF Gold
P: 39,556

For example, if M= 2, you can take i= <1, 0>, j= <0, 1>, and k= i+ j= <1, 1>. Then any subset of order 2, {i, j}, {i, k}, and {j, k}, is a basis.
For M= 3, start with i= <1, 0, 0>, j=<0, 1, 0>, and k= <0, 0, 1> and add l= i+ j+ k. Can you continue that? 


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