# Definition:Orthonormal Basis of Vector Space

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## Definition

Let $V$ be a vector space.

Let $\BB = \tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be a basis of $V$.

Then $\BB$ is an **orthonormal basis of $V$** if and only if:

- $(1): \quad \tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ is an orthogonal basis of $V$
- $(2): \quad \norm {\mathbf e_1} = \norm {\mathbf e_2} = \cdots = \norm {\mathbf e_n} = 1$

## Also known as

An **orthonormal basis** is also known as a **Cartesian basis**, particularly when it is used as the basis of a Cartesian coordinate system.

## Also see

- Results about
**orthonormal bases**can be found here.

## Sources

- 1992: Frederick W. Byron, Jr. and Robert W. Fuller:
*Mathematics of Classical and Quantum Physics*... (previous) ... (next): Volume One: Chapter $1$ Vectors in Classical Physics: $1.2$ The Resolution of a Vector into Components - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**orthonormal basis**