The following set of points is a lattice in RR^n: D_n= {(v_1,...,v_n)|v_1,...,v_n in bbbZ \ and \ v_1+...+v_n \ is \ even} Find vectors x_1,...,x_k such that D_n=Int(x_1,...,x_k)

Mbalisikerc

Mbalisikerc

Answered question

2022-07-14

The following set of points is a lattice in R n :
D n = ( v 1 , . . . , v n ) | v 1 , . . . , v n Z and v 1 + . . . + v n is even}
Find vectors x 1 , . . . , x k such that D n = I n t ( x 1 , . . . , x k )
My solution:
Firstly, I know that if x 1 , . . . , x k R n , then we define I n t ( x 1 , . . . , x k ) = t 1 x 1 + . . . + t k x k where t 1 , . . . , t k Z
So, I have deduced that the vectors x 1 , . . . , x k R n so that D n = I n t ( x 1 , . . . , x k ) are any vectors such that t 1 x 1 + . . . + t k x k = ( v 1 , . . . , v n ) where v 1 + . . . + v n are even.
However, I feel my solution is certainly lacking and I was wondering if anyone could help guide me in the right direction.

Answer & Explanation

encoplemt5

encoplemt5

Beginner2022-07-15Added 15 answers

D n is known as the checkerboard lattice and a basis for it is
x 1 = [ 1 1 0 0 ] , x 2 = [ 1 1 0 0 ] , x 3 = [ 0 1 1 0 0 ] , , x n = [ 0 0 1 1 ] .

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