10.1: Coordinate Vectors

Let \(V\) be a finite-dimensional inner product space with inner product \(\inner{\cdot}{\cdot}\) and dimension \(\dim(V)=n\). Then \(V\) has an orthonormal basis \(e=(e_1,\ldots,e_n)\), and, according to Theorem9.4.6~\ref{thm:ParsevalsIdentity}, every \(v\in V\) can be written as

\begin{equation*} v = \sum_{i=1}^n \inner{v}{e_i} e_i. \end{equation*}
This induces a map
\begin{equation*}
\begin{split}
[\,\cdot\,]_e : V &\to \mathbb{F}^n\\
v &\mapsto \begin{bmatrix}
\inner{v}{e_1}\\ \vdots \\ \inner{v}{e_n} \end{bmatrix},
\end{split}
\end{equation*}

which maps the vector \(v\in V\) to the \(n\times 1\) column vector of its coordinates with respect to the basis \(e\). The column vector \([v]_e\) is called the coordinate vector of \(v\) with respect to the basis \(e\).

Example \(\PageIndex{1}\):

Recall that the vector space \(\mathbb{R}_1[x]\) of polynomials over \(\mathbb{R}\) of degree at most 1 is an inner product space with inner product defined by

\begin{equation*}
\inner{f}{g} = \int_0^1 f(x)g(x)dx.
\end{equation*}
Then \(e=(1,\sqrt{3}(-1+2x))\) forms an orthonormal basis for \(\mathbb{R}_1[x]\). The coordinate vector of the polynomial \(p(x)=3x+2\in \mathbb{R}_1[x]\) is, e.g.,
\[ [p(x)]_e= \frac{1}{2} \begin{bmatrix} 7 \\ \sqrt{3} \end{bmatrix}. \]

Note also that the map \([\,\cdot\,]_e\) is an isomorphism (meaning that it is an injective and surjective linear map) and that it is also inner product preserving. Denote the usual inner product on \(\mathbb{F}^n\) by

\begin{equation*}
\inner{x}{y}_{\mathbb{F}^n} = \sum_{k=1}^n x_k \overline{y}_k.
\end{equation*}

Then

\begin{equation*}
\inner{v}{w}_V = \inner{[v]_e}{[w]_e}_{\mathbb{F}^n}, \qquad \text{for all \(v,w\in V\),}
\end{equation*}

since

\begin{multline*}
\inner{v}{w}_V = \sum_{i,j=1}^n \inner{\inner{v}{e_i} e_i}{\inner{w}{e_j}e_j}
= \sum_{i,j=1}^n \inner{v}{e_i} \overline{\inner{w}{e_j}} \inner{e_i}{e_j}\\
= \sum_{i,j=1}^n \inner{v}{e_i} \overline{\inner{w}{e_j}} \delta_{ij}
= \sum_{i=1}^n \inner{v}{e_i} \overline{\inner{w}{e_i}} = \inner{[v]_e}{[w]_e}_{\mathbb{F}^n}.
\end{multline*}

It is important to remember that the map \([\,\cdot\,]_e\) depends on the choice of basis \(e=(e_1,\ldots,e_n)\).

Contributors

• Isaiah Lankham, Mathematics Department at UC Davis
• Bruno Nachtergaele, Mathematics Department at UC Davis
• Anne Schilling, Mathematics Department at UC Davis

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