little linear algebra problem

Question

Suppose $G$ is a generator matrix (over $GF(q)$) of a linear $[n,k]$ code. A linear $[n,k]$ code is $k$-dimensional subspace of $(GF(q))^n$. A generator matrix $G$ is then a $k\times n$ matrix over $GF(q)$ such that its rows are a base of the code. Let $C(w)$ be a subset of the code and $<C(w)>$ be its linear span. Is there a (unique) smallest subset of the set of rows of $G$ with a linear span containing $<C(w)>$? At first, that seemed trivial to me, as the span of all the rows of G contains $<C(w)>$, so one could take the intersection (as one normally does). But then, does the linear span of that intersection still contains $<C(w)>$? So basically what I'm asking is this: given a finite dimensional vector space $V$ and a base $B$ of $V$, is there a smallest subset of $B$ with its linear span containing a given linear subspace $A$ of $V$? Well yes, there is, but is it - as my text in coding theory states - the intersection of all subsets of $B$ with their linear span containing $A$? I find that hard to believe.

Answer 1

Okay, I solved it (finally had the courage to actually think about it). The key thing is that base vectors are linearly independent. This insures that the span of the intersection of subsets of a base is equal to the intersection of their spans. For instance, if $r_1,r_2,r_3 \in B$ and $t_1,t_2,t_3,t_4 \in B$, with common element $b=t_3=r_2$ and if $\lambda_1t_1+\lambda_2t_2+\lambda_3t_3+\lambda_4t_4=\sigma_1r_1+\sigma_2r_2+\sigma_3r_3$ it follows from linear independence that $\lambda_3=\sigma_2$ and all the other scalars are zero.