Definition:Pseudoinverse of Linear Transformation

Definition
Let $U, V$ be vector spaces over a field $K$.

Let $S: U \to V$ be a linear transformation.

Let $T: V \to U$ be a linear transformation.

$S$ and $T$ are said pseudoinverse to each other :
 * $T \circ S - I_U$ is a degenerate linear operator on $U$
 * $S \circ T - I_V$ is a degenerate linear operator on $V$

where:
 * $\circ$ denotes the composition
 * $I_U$ denotes the identity mapping of $U$
 * $I_V$ denotes the identity mapping of $V$

Also see

 * Linear Transformation has Finite Index iff Pseudoinverse exists