Finding the orthogonal complement of a subspace of $\mathbb{R}^4$

Question

Let $V$ be the vector space $\{(x,y,z,w)\in\mathbb{R}^4:x+y-z=0\ \text{and}\ x+y+w=0\}$. Then, what would be a basis for the orthogonal complement of $V$.

I think we have to find the null space of the column space spanned by the above two vectors, i.e, the null space of $$\begin{pmatrix}1&1&0&0\\1&1&0&0\\2&0&0&0\\0&-2&0&0\end{pmatrix}$$ Am I right? Thanks beforehand.

Answer 1

The space $V$ is the space of those vectors $v\in\mathbb{R}^4$ such that $\bigl\langle v,(1,1,-1,0)\bigr\rangle=\bigl\langle v,(1,1,0,1)\bigr\rangle=0$. Since $(1,1,-1,0)$ and $(1,1,0,1)$ are linearly independent, they form the basis that you aer after.