Let $S$ be the subspace of $\mathbb{R}^4$ spanned by $x_1=(1,0,-2,1)^T$ and $x_2=(0,1,3,-2)^T$. Find a basis for $S_\perp$

Question

Let $S$ be the subspace of $\mathbb{R}^4$ spanned by $x_1=(1,0,-2,1)^T$ and $x_2=(0,1,3,-2)^T$. Find a basis for $S_\perp$.

For this kind of question, if the subspace is spanned by one vector, I know how to deal with it by setting a vector $Y=(y_1,y_2,y_3)$ such that $x^Ty=0$. But for this question, the subspace is spanned by two vectors, I don't know how to do it. I guess maybe I should multiply the two vectors and then set $y=(y_1,y_2,y_3)$ and
then calculate $x^Ty=0$ ?

Answer 1

First $y \in \mathbb{R}^4$, let $y = (y_1, y_2, y_3, y_4)^T$ rather than $y=(y_1, y_2, y_3)^T$.

We would want $y$ to satisfies.

$$x_1^Ty = 0$$ $$x_2^Ty = 0$$

That is

$$\begin{bmatrix} x_1^T \\ x_2^T\end{bmatrix}y = \begin{bmatrix} 0 \\ 0\end{bmatrix}$$