Link between adjoint operator $T^*$ and $T^t$

Question

There's this theorem in Hoffman and Kunze which states that if $T$ is a linear operator on $V$ then we see that matrix of $T^*$ in the orthogonal basis is the transpose of $T$(consider the filed to be as $\mathbb{R}$].My question is why does this happen ? Shouldn't it be finding out the cofactor and doing it ?What special properties does this orthonormal bais have?

Answer 1

Let $\{e_i\}$ be an orthonormal basis.

By definition of the matrix entries

$$T_{ij} = \langle Te_j , e_i \rangle$$ and by definition of $T^*$, for any two vectors $u,v$

$$\langle T^*u , v \rangle = \langle u , Tv \rangle$$ therefore

$$T^*_{ij}=\langle T^* e_j , e_i \rangle = \langle e_j , Te_i\rangle = \langle T e_i , e_j\rangle = T_{ji} = T^t_{ij}$$