Set of vectors with each subset forming a basis

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SUMMARY

The discussion focuses on constructing a set A consisting of N vectors of size M, where each subset S of A, with |S| = M, forms a basis for R^M. The user provides examples for M=2 and M=3, demonstrating that specific combinations of vectors can maintain linear independence. For M=2, the vectors <1, 0> and <0, 1> along with <1, 1> serve as a basis, while for M=3, the vectors <1, 0, 0>, <0, 1, 0>, <0, 0, 1>, and <1, 1, 1> are suggested. The discussion invites further exploration of explicit constructions or proofs of existence for larger sets.

PREREQUISITES
  • Understanding of linear independence in vector spaces
  • Familiarity with the concept of bases in R^M
  • Knowledge of vector addition and scalar multiplication
  • Basic principles of combinatorial mathematics
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  • Explore the concept of vector spaces in linear algebra
  • Research the properties of linear combinations and their implications
  • Learn about the construction of bases for higher-dimensional spaces
  • Investigate combinatorial methods for selecting linearly independent sets
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Mathematicians, students of linear algebra, and anyone interested in vector space theory and the construction of bases in higher dimensions.

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