= Dual image and kernel annihilator identity
{title2=$\operatorname{im}(\alpha^\vee)=(\ker\alpha)^\circ$}
For a <linear map> between finite-dimensional <vector spaces>, the image of its <dual map> is the <annihilator of a vector subspace> given by its kernel. Inclusion follows by composition. Conversely a <linear functional> vanishing on the kernel descends to the image and extends to the target by <basis extension>. The identity and the <rank-nullity theorem> show that a linear map and its dual have equal rank.
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