Definition:Orthonormal Basis of Vector Space

From ProofWiki
Jump to navigation Jump to search


It has been suggested that this page be renamed.
In particular: Definition:Orthonormal Basis of Finite Dimensional Vector Space
To discuss this page in more detail, feel free to use the talk page.


Definition

Let $\struct {V, \norm {\, \cdot \,} }$ be a normed vector space.

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


Then $\BB$ is an orthonormal basis of $\struct {V, \norm {\, \cdot \,} }$ 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

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Orthonormal_Basis_of_Vector_Space&oldid=629216"