Let $X$, $Y$ be finite dimensional inner product spaces, let $A: X \to Y$ be a linear operator, let $A^*: Y \to X$ be the adjoint operator to the linear operator, defined using $<y, Ax>_Y = <A^*y, x>_X$, $x \in X, y \in Y$ .
We know basic fact that $X,Y$ are partitioned into orthogonal subspaces: Nullspace of A and Range of the Adjoint, so we can come up with the following diagram:
Can someone check that I have the mapping defined correctly via the above diagram? Is there something I could do that distinguish the operators that map an element to $0$ from the operators that map element to the range?
Is there anything I can do to improve the diagram?