True or False:
If $\{V_1,...,V_n\}$ is a basis of $\Bbb R^n$ and $S = \text{span}\ \{V_1,\ldots,V_k\}$ for some positive integer $k<n$, then $S^\perp = \text{span}\{V_{k+1},\ldots, V_n\}$.
Prove true or give a counterexample if false. I would appreciate any sort of hint on about doing this question.
$$V= \text{span}\{(1,1),(1,0)\}, \qquad k = 1 < 2 = n$$ then $$S = \{(t,t): t \in \Bbb R\}$$ and $$S^\perp = \{(t,-t): t \in \Bbb R\} \neq \{(t,0): t \in \Bbb R\} = \text{span}\{(1,0)\}$$