= Solution
{c}
The <annihilator of a vector subspace> consists of the <linear functionals> that vanish on every element of that <vector subspace>:
$$
\boxed{W^0=\{\lambda\in V':\lambda(w)=0\text{ for every }w\in W\},\qquad
\dim W^0=\dim V-\dim W.}
$$
For the <dimension> identity, choose a <basis> $w_1,\ldots,w_k$ of $W$ and use <basis extension> to obtain $w_1,\ldots,w_n$ for $V$. A <linear functional> is determined by its values on these $n$ <basis vectors>. Belonging to the <annihilator of a vector subspace> sets its first $k$ values to zero and leaves its last $n-k$ values arbitrary. Equivalently the last $n-k$ elements of the <dual basis> form a <basis> of $W^0$.
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