I need some help understanding this solution to a problem. I am working on the problem above. I know that in order for a set of vectors to be a basis it must be linearly independent and span the given space. In the solution above it says that $v_1$ and $v_2$ will not be in the span for $H$. Doesn't the solution $0v_1+v_2+0v_3$ span $H$? Why does it say that $v_1$ and $v_2$ are not in the span?
I believe what was meant is that $v_1$ and $v_3$ aren't in $H$.
As $v_1, v_3 \not\in H$ and $v_1, v_3 \in \operatorname{span}\{v_1, v_2, v_3\}$, we see that $H \neq \operatorname{span}\{v_1, v_2, v_3\}$, so $\{v_1, v_2, v_3\}$ is not a spanning set for $H$.
You claimed that $\{v_2\}$ spans $H$, but this is false as $(1, 0, 0) \in H$ but $(1, 0, 0) \notin \operatorname{span}\{v_2\}$. Even if $H$ were spanned by $v_2$, then as $v_2 \in \operatorname{span}\{v_1, v_2, v_3\}$, we could only say that $H \subseteq \operatorname{span}\{v_1, v_2, v_3\}$.