#🔒 Algebra algorithm

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trim carbon
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I am writing an algorithm that generates all vector of each basis of the vector space Z2^n over Z2, but something isn't working properly. I know that for checking if a vector is linearly independent I verify the matrix rank, but I am not really sure

lost ferryBOT
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@trim carbon

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trim carbon
# 
import numpy
from itertools import combinations
from itertools import permutations

n = int(input())
p = 1

# formula for calculating number of bases of vector space Z2^2 over Z2
# for the first vector we have 2^n - 1 options (it can't be null)
# for the second vector we have 2^n - 2 options (it can't be null or the same as the first)
# for the third vector we have 2^n - 2^2 options (it can't be null or a linear combination of the first two)
# and so on
for i in range(n):
    p = p * (2 ** n - 2 ** i)

print("1)", p)

if n == 1:
    print("2) (1)")
elif n <= 4:
    print("2)")
    vector_list = []
    bases = []

    # generating the initial vectors using bit operations
    for i in range(2 ** n):
        vector = [(i >> j) & 1 for j in range(n)]
        vector_list.append(tuple(vector))

    count = 1
    for combination in combinations(vector_list, n):  # we get every combination of the initial vectors
        matrix = numpy.array(combination)
        if numpy.linalg.matrix_rank(
                matrix) == n:  # we check it's linear independency by calculating the rank of the matrix formed by those vectors
            for permutation in permutations(combination):  # we print every permutation of a valid basis
                print(f"{count}:", permutation)
                count += 1
    if n == 4:
        print("1)", p)  # it may not show at the start because of the huge number of vectors
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here is the code

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and

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for n = 3

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it should give me 168 vectors but it gives me 174

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here is the list

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should i verify something else?

lost ferryBOT
#

@trim carbon

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