Hi guys. Let $L\colon V\to V$ be a linear transformation of the inner product space $(V,\mathbb C,\langle{.,.}\rangle)$. Let $\beta=\langle b_1,\dots,b_n\rangle$ be an orthonormal basis of $V$. My book says that if $L(b_j)=\alpha_1b_1+\dots\alpha_nb_n$, then
$$
e_i^Tco_\beta(L(b_j))=\alpha_i=\langle b_i,L(b_j)\rangle.
$$
I don’t understand why we can also say that $\langle b_i,L(b_j)\rangle=\alpha_i$. The inner product of $V$ hasn't been defined, so how do we know that it will be multiplied terms wise?