Say there are $n$ vectors $a_1,\ldots,a_n$ in $\mathbb{R}^{d}$. What is the maximum number of such vectors I can have such that they are linearly independent and their first $k$ entries sum to zero?
Without the sum constraint, the answer is of course $d$. I can see the answer for some simple cases with $d=3$, $k=1,2$ etc., but I can't think of a way to approach this in general. Any hints?
Consider the set of vectors consisting of the vector with the first $k$ coordinates $1$ (and the remaining components $0$), the vectors with $1$ as first component and $-1$ as $i$th component for $2\leq i\leq k$ (otherwise $0$), and the vectors with only a $1$ as the $j$th component for $j>k$. This set is a basis for $\Bbb R^d$.
Say you have a vector where the sum of the first $k$ components is zero (in the standard basis). What property does that vector have in the basis described above? How many linearly independent such vectors can you have?