$U_k = \{(x_i)_{1≤i≤n} \in \mathbb{R}^n\ |\ x_k = 0\}$. Is this a vector subspace of $\mathbb{R}^n$? For $n = 3$, what is the smallest vector subspace of $\mathbb{R}^n$ that contains $U_1, U_2, U_3$.
Apparently only $2$ vectors are needed to span this. And the smallest subspace is $\mathbb{R}^3$ itself. What does that mean?
We can prove directly that $U_k$ is a subspace of $\Bbb R^n$ or use that $U_k$ is the kernel of the linear form $$\varphi_k\colon \Bbb R^n\rightarrow \Bbb R,\; (x_i)_{1\le i\le n}\mapsto x_k$$
Notice that $U_k$ is an hyper-plan and the smallest subspace that contains two different hyper-plans is the whole space.