I had some difficulties with this quite difficult problem.
The vector space is $\Bbb Z_{2}^n$, we have
$(1,1,0,\dots,0)$, $(0,1,1,0,\dots,0)$, $\dots$, $(0,\dots,0,1,1)$, $(1,0,\dots,1)$.
What is the rank of this vector space?
Thank you very much for your help, best wishes,
Tom
Assume there is a linear combination of the first $n-1$ of these vectors that is equal to $0$. Since these are vectors over $\Bbb Z_2$, the coefficients are either $1$ or $0$. We have $$ a_1(1, 1, 0,\ldots,0) + a_2(0,1,1,0,\ldots,0) + \cdots +a_{n-1}(0, \ldots,0,1,1) = (0,0,\ldots,0) $$ Since the first vector is the only one that has anything in the first coordinate, we must have $a_1 = 0$. But then, the second vector is the only one that has any contribution to the second coordinate, so $a_2$ also has to be $0$. And so on. Therefore there is no non-trivial linear combination of the first $n-1$ of these vectors that equals the zero vector, and your set of vectors has rank at least $n-1$.
Does it have rank $n$? What is the sum of all the $n$ vectors?