Why is the name "orthogonality"?

Let $G$ be a finite abelian group and $\hat G$ be the character group of $G$.Let $G=\{a_1,a_2,...,a_n\}$ and $\hat G=\{f_1,f_2,...,f_n\}$.$a_1$ is the identity of $G$ and $f_1$ is the principal character.Then we have the following relations:

$\sum\limits_{i=1}^n f_i(a_j)=\begin{cases}n &\text{if }j=1\\0&\text{otherwise} \end{cases}$

and,

$\sum\limits_{j=1}^n f_i(a_j)=\begin{cases}n &\text{if }i=1\\0&\text{otherwise} \end{cases}$

I want to know why these are called "orthogonality" relation.I want to find suitable reason for nomenclature.

The definition of orthogonal which we know is something like $f_i(a_j)=\begin{cases} 1 &\text{if } i=j\\0&\text{otherwise}\end{cases}$

But here it is not so.So can someone explain me the reason behind the name "orthogonality".

Solution 1:

These relations say that the characters of $G$ are orthogonal for the inner product $\langle .,. \rangle$ defined on $\mathbb C^{G}$ (the space of all functions from $G$ to $\mathbb{C}$, which has dimension $n$) by $$\langle \varphi, \psi \rangle = \frac{1}{n} \sum_{g \in G} \overline{\varphi(g)}\psi(g)$$