27 prove that if b1bn1 are q linearly independent then for every e 0 there exists z1zn1 e z such

Name: 27 prove that if b1bn1 are q linearly independent then for every e 0 there exists z1zn1 e z such that 0 lz1b1 zn1bn1l ie if the b do not span a lattice then their integer span contains arbit 83124
Uploaded: 2023-02-03T21:27:07-08:00
Duration: 5 min 18 s
Channel: Victoria Chayes
Description: 27 prove that if b1bn1 are q linearly independent then for every e 0 there exists z1zn1 e z such that 0 lz1b1 zn1bn1l ie if the b do not span a lattice then their integer span contains arbit 83124

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Victoria Chayes · Accepted Answer

First, let&#x27;s assume that the vectors b1, ..., bn+1 are Q-linearly independent. This means that t

Question

2.7 Prove that if b1,...,bn+1 are Q-linearly independent, then for every e > 0 there exists z1,...,zn+1 âˆˆ Z such that 0 < |z1b1 +...+ zn+1bn+1| < Îµ. (I.e., if the bi do not span a lattice, then their integer span contains arbitrarily short vectors.)

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