orthonormal basis — Jemoka Knowledge Base

An Orthonormal basis is defined as a basis of a finite-dimensional vector space that’s orthonormal. Additional Information orthonormal list of the right length is a basis An orthonormal list is linearly independent, and linearly independent list of length dim V are a basis of V. \blacksquare Writing a vector as a linear combination of orthonormal basis According to Axler, this result is why there’s so much hoopla about orthonormal basis. Result and Motivation For any basis of V, and a vector v \in V, we by basis spanning have:

\begin{equation} v = a_1e_1 + \dots a_{n}e_{n} \end{equation}

Yet, for orthonormal basis, we can actually very easily know what the a_{j} are (and not just that some a_{j} exist). Specifically:

\begin{equation} a_{j} = \langle v,e_{j} \rangle \end{equation}

That is, for orthonormal basis e_{j} of V, we have that:

\begin{equation} v = \langle v, e_{1} \rangle e_{1} + \dots + \langle v, e_{n} \rangle e_{n} \end{equation}

for all v \in V. Furthermore:

\begin{equation} \|v\|^{2} = | \langle v,e_1 \rangle|^{2} + \dots + | \langle v, e_{n} \rangle|^{2} \end{equation}

Proof Given e_{j} are basis (nevermind orthonormal quite yet), we have that:

\begin{equation} v = a_1e_{1} + \dots + a_{n}e_{n} \end{equation}

WLOG let’s take \langle v, e_{j} \rangle:

\begin{equation} \langle v,e_{j} \rangle = \langle a_1e_1 + \dots +a_{n}e_{n}, e_{j} \rangle \end{equation}

Given additivity and homogenity in the first slot, we now have:

\begin{equation} \langle v, e_{j} \rangle = a_{1}\langle e_1, e_{j} \rangle + \dots +a_{n}\langle e_{n}, e_{j} \rangle \end{equation}

Of course, each e_{i} and e_{j} are orthogonal, so for the most part a_{i}\langle e_{i}, e_{j} \rangle = 0 for i \neq j. Except where a_{j} \langle e_{j}, e_{j} \rangle = a_{j} 1 = a_{j} because the e vectors are also norm 1. Therefore:

\begin{equation} \langle v, e_{j} \rangle= 0 + \dots +a_{j} + \dots +0 = a_{j} \end{equation}

We now have \langle v,e_{j} \rangle = a_{j} WLOG for all j, as desired. Plugging this in for each a_{j} and applying Norm of an Orthogonal Linear Combination yields the \|v\|^{2} equation above. \blacksquare