Why, when $m < n$, does the vector space $S$ of vectors $x ∈ \mathbb{R}^n$ such that $a_i · x = 0$ always contains a nonzero vector? Fix vectors $a_1, . . . , a_m ∈ \mathbb{R}^n$, and let S be the set of vectors $x ∈ \mathbb{R}^n$ such that $a_i · x = 0$ for all $i$.
Now I know that $S$ is a vector space. Why is it that when $m < n$, the vector space $S$ always contains a nonzero vector?
My idea is to use induction. If $x = (x_1, . . . , x_n)$, I want to solve one of the equations $a_i · x = 0$ for $x_n$ in terms of $(x_1, . . . , x_{n−1})$ and substitute the result into the other equations to get a system of one fewer equations in one less unknown.
I'm just not sure how to write this out or perform this induction. Can someone show me how to start doing this?

Suppose $S=\{0\}$. Then $dim(S)=0$. Thus $\dim(S^\perp)=n$, so $S^\perp=\mathbb{R}^n$. But $\mathrm{span}(\{v_1,...,v_m\})=S^\perp$ which implies $n=\mathrm{dim(S^\perp)}=\mathrm{dim}(\mathrm{span}(\{v_1,...,v_m\}))\le m<n$. This contradiction shows $S\neq\{0\}$
