Rank and Nullity of Transpose

Theorem
Let $G$ and $H$ be $n$-dimensional vector spaces over a field.

Let $\map \LL {G, H}$ be the set of all linear transformations from $G$ to $H$.

Let $u \in \map \LL {G, H}$.

Let $u^\intercal$ be the transpose of $u$.

Then:
 * $u$ and $u^\intercal$ have the same rank and nullity

Proof
From Rank Plus Nullity Theorem and Results Concerning Annihilator of Vector Subspace:

Hence it follows that $u$ and $u^\intercal$ have the same rank and nullity.