\[\newcommand{\u}[1]{\boldsymbol{\mathsf{#1}}}
\renewcommand{\b}[1]{\boldsymbol{#1}}
\newcommand{\t}[1]{\textsf{#1}}
\newcommand{\m}[1]{\mathbb{#1}}
\def\RR{\bf R}
\def\bold#1{\bf #1}
\def\mbf#1{\mathbf #1}
\def\uv#1{\hat{\usf {#1}}}
\def\dl#1{\underline{\underline{#1}}}
\newcommand{\usf}[1]{\boldsymbol{\mathsf{#1}}}
\def\bs#1{\usf #1}\]
Maps between Euclidean spaces
Definition The map \(T:\mathbb{E}^{N} \to \mathbb{F}^{N}\) assigns to each vector \(\boldsymbol{v}\) in \(\mathbb{E}^{N}\) the unique vector \(T(\boldsymbol{v})\) in $\mathbb{F}^{N}$. It is called a map between Euclidean spaces.
Examples of maps between Euclidean spaces
Let $(\boldsymbol{e}_i)$ be an orthonormal basis of $\mathbb{E}^N$, and $(\boldsymbol{f}_i)$ be an orthornormal basis for $\mathbb{F}^N$.
Let $\boldsymbol{v} \in \mathbb{E}^N$.
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\[\boldsymbol{T}_{1}(\boldsymbol{v})=(\boldsymbol{v}\cdot \boldsymbol{e}_i)\boldsymbol{f}_i\]
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\[\boldsymbol{T}_{2}(\boldsymbol{v})=\boldsymbol{T}_1(\boldsymbol{v})/\lVert \boldsymbol{T}_1(\boldsymbol{v}) \rVert\]
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\[\boldsymbol{T}_{3}(\boldsymbol{v})=\boldsymbol{0}_{\mathbb{F}^{N}}\]