Dual map

The dual of $\alpha :V\to W$ is ${\alpha }^{\ast }:W^{\ast }\to V^{\ast }$ defined by ${\alpha }^{\ast }(g)=g\circ \alpha$. In finite dimensions,

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.

For the map represented by a matrix $A$, column rank is the rank of the map and row rank is the rank of its dual. Since the dual map has kernel equal to the annihilator of the original image, rank-nullity gives equal ranks.

If the columns $v_1,\dots ,v_m$ of an $n$ by $m$ matrix are independent, some $m$ by $m$ row minor is nonzero. A nonzero term of that minor's determinant selects distinct nonzero coordinates $f(i)$, and replacing the corresponding standard basis vectors $e_{f(i)}$ by the $v_i$ leaves a basis.

For a finite-dimensional vector space and $U\le V$, dualizing the quotient and inclusion maps gives