= Solution
{c}
Define the <linear functional> $L:F^n\to F$ by $L(x)=\sum_{i=1}^n a_ix_i$. The given <hyperplane> is $H=\ker L$. Choose $j$ with $a_j\ne0$. For any $t\in F$, the <vector> whose only nonzero coordinate is $x_j=t/a_j$ satisfies $L(x)=t$. Hence $L$ is <surjective> and has <rank of a linear map> equal to $\dim_F F=1$.
Applying the <rank-nullity theorem> to this particular <linear map> gives
$$
\boxed{\dim_F H=n-1.}
$$
The nonzero-<coefficient> hypothesis is essential: if all $a_i$ were zero, the <kernel of a linear map> would instead be all of $F^n$.
Back to article page
