Say, we have a vector space $V$ (assume dim $>=3$), over the field of rational numbers with the basis $v_1...v_n$. Now, if we define the linear functionals $g_i$ as $g_i(v_j) = j+(i-1)*(-1)^{ij}$, for $i,j\in[1,n]$, what is the dim of span{$g_1,....,g_n$}?
I know the key is to find out the LI element number among such $g_i$, however, are there any techniques on such $g_i$?
And by the way, what is the intersection of those $g_i$'s Kernels, aka is there some basis $v_i$ such that for all $g_i$ we have $g_i=0$?
(Promoting my comment to an answer.)
Since you have finite dimensions, there is some row-vector representation of each $g_i$. (There is a theorem for this, but I forgot the name of it.)
You can initially assume that $V$ has the standard basis to determine the dimension of the span of the functionals, since a change in basis won't change the dimension. In other words, start by stuffing $g_i(v_j)$ into a row vector to represent $g_i$. This will be sufficient to determine linear independence of $\{g_i\}_{i=0}^n$.