<s>[INST] Let $\textbf{v} = (1, 1, 1, 1)$. Find a basis for... Let $\textbf{v}=(1,1,1,1)$. Find a basis for $\{\textbf{u}\in\Bbb{R}^4\ |\ \textbf{u}\cdot\textbf{v}=0\}$
How can I do this? In particular, I do not understand $\textbf{u}\cdot\textbf{v}=0.$
 [/INST] Denote $W=\{u\in\mathbb{R}^4\mid u\cdot v=0\}$, we claim that the set
$\beta=\left\{
\begin{pmatrix}-1\\1\\0\\0\end{pmatrix},
\begin{pmatrix}-1\\0\\1\\0\end{pmatrix},\begin{pmatrix}-1\\0\\0\\1\end{pmatrix}
\right\}$ is a basis for $W$. First, given 
$u=(u_1,u_2,u_3,u_4)\in W$, then we have
$$0=u\cdot v=
\begin{pmatrix}u_1\\u_2\\u_3\\u_4\end{pmatrix}\cdot
\begin{pmatrix}1\\1\\1\\1\end{pmatrix}=
u_1+u_2+u_3+u_4,$$ that is, $u_1=-u_2-u_3-u_4$. So we can write
$$u=\begin{pmatrix}-u_2-u_3-u_4\\u_2\\u_3\\u_4\end{pmatrix}
=u_2\begin{pmatrix}-1\\1\\0\\0\end{pmatrix}
+u_3\begin{pmatrix}-1\\0\\1\\0\end{pmatrix}
+u_4\begin{pmatrix}-1\\0\\0\\1\end{pmatrix},$$
which follows that $\beta$ spans $W$. Next, we show that $\beta$ is linearly independent, but it is easy to prove so I leave it to you. Hence we conclude that
$\beta$ is a basis for $W$, and the claim is proved.
</s>