In one lectures on linear algebra, my professor wrote on the first day:
Let $L$ be in a linear map $L: X \to Y$, then $Im(L) \oplus Ker(L^*) = Y$
On the second day, he wrote: $L: X \to Y$, then $Im(L) \oplus Ker(L^T) = Y$
I am a little bit confused by the mixture of adjoint and transpose. I suspect the transpose is used by assuming that $L$ is the matrix representation of the transform, whereas adjoint is just a general symbol to denote the adjoint of a linear transform and does not induce a transpose.
Can someone clarify this concept for me? THanks!