How to find a subspace with maximum dimension that doesn't include $k$ special vectors?

Question

Assume $V_n = \{(a_1,a_2,\cdots,a_n), a_i \in \mathbb{GF}(2)\}$ and $k$ linearly independent vectors of $V_n$. How to find a subspace with maximum dimension that doesn't include these $k$ vectors?

Answer 1

The dimension is always $\;n-1\;$ as we can always find a hyperplane not containing any of the $\;k\;,\;k<n\;$ , lin. ind. vectors , say $\;\{v_1,...,v_k\}\;$.

Proof: First, complete the above set to a basis of $\;V_n\;$: $\;\{v_1,...,v_k,v_{k+1},...,v_n\}\;$ , and now define

$$\phi\in V_n^*\;,\;\;\phi(v_i)=\begin{cases}1&,\;\;i=1,2,...,k\\{}\\0&,\;\;i=k+1,...,n\end{cases}$$

Clearly $\;\phi\;$ is a non-zero linear functional and as such its kernel is a hyperplane (i.e., maximal subspace), and it contains none of $\;v_1,...,v_k\;$

Note that the definition field is irrelevant here.