Coordinates in an orthonormal basis

Statement

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, and let $(e_1, \dots, e_n)$ be an orthonormal basis of $V$. Then, for each $v \in V$, $$v = \sum_{i=1}^{n} \langle v, e_i \rangle\, e_i.$$ So the coordinates of $v$ in this basis are the inner products $\langle v, e_i \rangle$.

Remarks

If $v = \sum_j a_j e_j$, then $\langle v, e_i \rangle = \sum_j a_j \langle e_j, e_i \rangle = a_i$. For the basis $\frac{1}{\sqrt 2}(1, 1)$, $\frac{1}{\sqrt 2}(1, -1)$ of $\R^2$, the vector $(3, 1)$ has the coordinates $4/\sqrt 2$ and $2/\sqrt 2$. With the convention of linearity in the second argument, the coefficient is written $\langle e_i, v \rangle$.

Explanations

L1 human

In an orthonormal basis, we find each coordinate of a vector with one inner product. This is much easier than solving a linear system. It is the finite-dimensional form of a Fourier expansion.