Is there any way to find an explicit formula for the adjoint of a linear transformation?

Question

I know that the definition of the adjoint of a linear transformation is defined to be $\langle T(x), y \rangle = \langle x, T^{*}(y) \rangle$ but is there any way to find an explicit formula for $T^{*}(y)$ if you know $T$ and $y$?

Answer 1

If the vector space is finite-dimensional and $A$ is the matrix representation of $T$ with respect to the standard basis, then the matrix of representation of $T^*$ is given by conjugating the entries of $A$, then taking the transpose. This is generally denoted $A^*$ or $A^H$.