Is $\{x \in K^n | x_1x_5=1 \} \text{ a } K \text{-vector subspace of } K^n \text{?}$
I think it is not because
$ \left( \begin{array}{c} 5 \\ 6 \\ 7 \\ 1 \\ \frac {1}{5} \end{array} \right) + \left( \begin{array}{c} 5 \\ 6 \\ 7 \\ 1 \\ \frac{1}{5} \end{array} \right) = \left( \begin{array}{c} 10 \\ 12 \\ 14 \\ 2 \\ \frac{4}{10} \end{array} \right) \ $ $\notin \{x \in K^n | x_1x_5=1 \}$
Question: Is that enough to show that the given set is not a $K$-vector subspace of $K^n$ or do I miss something here?