Definition:Orthonormal Basis of Vector Space

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

• Definition:Cartesian Coordinate System

• 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