An orthonormal family is linearly independent

Statement

Let $V$ be an inner product space over $\R$ or $\C$. Then each orthonormal family in $V$ is linearly independent.

Remarks

If $\sum_i a_i e_i = 0$, then the inner product with $e_j$ gives $a_j = 0$. More generally, a family of nonzero pairwise orthogonal vectors is linearly independent. So an orthonormal family in an $n$-dimensional space has at most $n$ vectors.

Explanations

L1 human

Vectors at right angles cannot be combinations of each other. To find the coefficient of $e_j$ in a combination, take the inner product with $e_j$. If the combination is zero, then every coefficient is zero.