Denote $Z$ as the set of points in $\mathbb{R}^n$ whose coordinates are $0$ or $1$. Find the maximum, of the number of points in $Z \cap V$. Denote $Z$ as the set of points in $\mathbb{R}^n$ whose coordinates are $0$ or $1$. Let $k$ be given, $0 \leq k \leq n$. Find the maximum, over all vector subspaces $V \subset \mathbb{R}^n$ of dimension $k$, of the number of points in $Z \cap V$.

My attempt: Since $\dim(V)=k$, it has $k$ linear independent vectors. Thus, the maximum number of points is $2^k$. 

Is my proof correct? It seems a bit too easy. 

Remark: The question is taken from the book 'Putnam and Beyond', page $78$.
we do have $2^n$ of these points one of them is the 0-vector. There are $2^{n-1}$ linearly independent vectors. If V has dimension k then it can have at most k of these vectors plus the 0-vector and this adds up to k+1.
we do have $2^n$ of these points one of them is the 0-vector. There are $2^{n-1}$ linearly independent vectors. If V has dimension k then it can have at most k of these vectors plus the 0-vector and this adds up to k+1.