Set of vectors with each subset forming a basis


by Constantinos
Tags: basis, forming, subset, vectors
Constantinos
Constantinos is offline
#1
Nov24-12, 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!
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Vargo
Vargo is offline
#2
Nov26-12, 01:00 PM
P: 350
Do you want an explicit construction or a proof that such a set exists?
HallsofIvy
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#3
Nov27-12, 08:37 AM
Math
Emeritus
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Thanks
PF Gold
P: 38,894
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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