\(\DeclarePairedDelimiterX{\Set}[2]{\{}{\}}{#1 \nonscript\;\delimsize\vert\nonscript\; #2}\) \( \DeclarePairedDelimiter{\set}{\{}{\}}\) \( \DeclarePairedDelimiter{\parens}{\left(}{\right)}\) \(\DeclarePairedDelimiterX{\innerproduct}[1]{\langle}{\rangle}{#1}\) \(\newcommand{\ip}[1]{\innerproduct{#1}}\) \(\newcommand{\bmat}[1]{\left[\hspace{2.0pt}\begin{matrix}#1\end{matrix}\hspace{2.0pt}\right]}\) \(\newcommand{\barray}[1]{\left[\hspace{2.0pt}\begin{matrix}#1\end{matrix}\hspace{2.0pt}\right]}\) \(\newcommand{\mat}[1]{\begin{matrix}#1\end{matrix}}\) \(\newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}}\) \(\newcommand{\mathword}[1]{\mathop{\textup{#1}}}\)
Needs:
Real Inner Product ›
Independent Set of Real Vectors ›
Real Vector Angles ›
Needed by:
Orthogonal Real Matrices ›
Orthonormal Set of Vectors ›
Real Matrices with Orthonormal Columns ›
Links:
Sheet PDF ›
Graph PDF ›

Orthonormal Set of Real Vectors

Definition

A vector is normalized if its norm is 1. A set of vectors $\set{u_1, \dots , u_k}$ orthogonal if $u_i^\top u_j = 0$ whenever $i \neq j$. A set of vectors is orthonormal if the set is orthogonal and each vector is normalized.

Basis

An orthonormal set of vectors is also an independent set. In other words, orthonormality is stronger than independence.

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