Orthonormal basis

Definition

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$. An orthonormal basis of $V$ is a basis $(e_1, \dots, e_n)$ of $V$ that is an orthonormal family.

Remarks

An orthonormal family of $\dim V$ vectors is an orthonormal basis. In an orthonormal basis, coordinates are inner products: $v = \sum_i \langle v, e_i \rangle e_i$. The vectors $\frac{1}{\sqrt 2}(1, 1)$ and $\frac{1}{\sqrt 2}(1, -1)$ form an orthonormal basis of $\R^2$. In infinite-dimensional Hilbert spaces, the term means a maximal orthonormal family, which is in general not a basis in the algebraic sense.

Explanations

L1 human

An orthonormal basis is a set of perpendicular unit vectors that spans the space. Each coordinate of a vector is its inner product with one basis vector. No linear system needs to be solved to find coordinates.