How do I prove that a subspace of a vector space $X$ is the null space of some linear functional on $X$?

Question

How do I prove that a subspace of a vector space $X$ is the null space of some linear functional on $X$? Say $\dim(X) = n$ and $Z$ is the subspace and $\dim(Z)=n-1$. Further, how to show that the functional is always uniquely determined to within a scalar multiple?

I'm having a difficult time coming up with a functional $f$ and don't understand how I would have to go about proving it unique to within a scalar multiple. Any help will be greatly appreciated.

Answer 1

Choose a basis of $Z$, say $\{e_i\}$. By extension lemmas, we can find a $v$, such that $\{e_i, v\}$ is a basis of $X$. Then we can define a linear functional $f:X\to k$ by $f(v)=f(\sum_i a_ie_i+a_nv)=a_n$. Moreover any functional that kills $Z$ has $f(\sum_i a_ie_i+a_nv)=f(v)a_n$, which shows that it differs from $f$ by $f(v)$, which shows uniqueness.