Dual image and kernel annihilator identity ($\mathrm{im}({\alpha }^{\vee })=(\ker \alpha {)}^{\circ }$)

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.