# 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.