\[\newcommand{\u}[1]{\boldsymbol{\mathsf{#1}}}
\renewcommand{\b}[1]{\boldsymbol{#1}}
\newcommand{\t}[1]{\textsf{#1}}
\newcommand{\m}[1]{\mathbb{#1}}
\def\RR{\bf R}
\def\bold#1{\bf #1}
\def\mbf#1{\mathbf #1}
\def\uv#1{\hat{\usf {#1}}}
\def\dl#1{\underline{\underline{#1}}}
\newcommand{\usf}[1]{\boldsymbol{\mathsf{#1}}}
\def\bs#1{\usf #1}\]
Linear mapping between Euclidean spaces
Definition: A mapping \(T : \mathbb{E}^N \to \mathbb{F}^M\) is called a linear mapping between Euclidean spaces iff for all \(\boldsymbol{a}\), and \(\boldsymbol{b}\in \mathbb{E}^N\) and for all \(\alpha \in \mathbb{R}\) we have
\(\begin{align}
\boldsymbol{T}(\boldsymbol{a}+_{\mathbb{E}^N}\boldsymbol{b})&=\boldsymbol{T}(\boldsymbol{a})+_{\mathbb{F}^M}\boldsymbol{T}(\boldsymbol{b})\\
\boldsymbol{T}(\alpha\cdot_{\mathbb{E}^N}\, \boldsymbol{a})&= \alpha\cdot_{\mathbb{F}^M} \boldsymbol{T}(\boldsymbol{a})
\end{align}\)
(linear) operator
Definition: We will refer to a linear mapping between Euclidean spaces as a (linear) operator.
Notation: We denote the set of all linear operators between $\mathbb{E}^N$ and $\mathbb{F}^M$ as $\mathcal{L}(\mathbb{E}^N,\mathbb{F}^M; \mathbb{R})$.
Examples and non-examples of operators
Let $(\boldsymbol{e}_i)$ and $(\boldsymbol{f}_i)$, respectively, be bases for $\mathbb{E}^N$ and $\mathbb{F}^N$. Let $\boldsymbol{I}: \mathbb{E}^N \to \mathbb{F}^N $ be defined as $I(\boldsymbol{e}_i)=\boldsymbol{f}_i$, and $\boldsymbol{v} \in \mathbb{E}^N$. Let $\boldsymbol{T}_i:\mathbb{E}^N\to \mathbb{F}^N$, $i=1,\ldots,5$.
- \(\boldsymbol{T}_{1}(\boldsymbol{v})=\boldsymbol{I}\boldsymbol{v}/2\), Yes
- \(\boldsymbol{T}_{2}(\boldsymbol{v})=(\boldsymbol{v}\cdot \boldsymbol{v})\, \boldsymbol{I}\boldsymbol{v}\), No
- \(\boldsymbol{T}_{3}(\boldsymbol{v})=\boldsymbol{0}_{\mathbb{F}^N}\),Yes
- \(\boldsymbol{T}_{4}(\boldsymbol{v})=\boldsymbol{k}\), where \(\boldsymbol{k}\in \mathbb{F}^N\neq 0_{\mathbb{F}^N}\), No
- \(\boldsymbol{T}_{5}(\boldsymbol{v})=\boldsymbol{ k }(\boldsymbol{b}\cdot_{\mathbb{E}^N} \boldsymbol{v})\), where where \(\boldsymbol{k}\in \mathbb{F}^N\), and where \(\boldsymbol{b}\in \mathbb{E}^N\), Yes
Mathematical Tensors
Definition Tensors are special types of linear operators in which \(\mathbb{F}^M=\mathbb{E}^N\)
Examples and non-examples of tensors
Let $\boldsymbol{T}_i:\mathbb{E}^N\to \mathbb{E}^N$, $i=1,\ldots,5$.
Let $\boldsymbol{v} \in \mathbb{E}^N$.
- \(\boldsymbol{T}_{1}(\boldsymbol{v})=\boldsymbol{v}/2\), Yes
- \(\boldsymbol{T}_{2}(\boldsymbol{v})=(\boldsymbol{v}\cdot \boldsymbol{v})\, \boldsymbol{v}\), No
- \(\boldsymbol{T}_{3}(\boldsymbol{v})=\boldsymbol{0}_{\mathbb{E}^N}\),Yes
- \(\boldsymbol{T}_{4}(\boldsymbol{v})=\boldsymbol{k}\), where \(\boldsymbol{k}\in \mathbb{E}^N\neq 0_{\mathbb{E}^N}\), No
- \(\boldsymbol{T}_{5}(\boldsymbol{v})=\boldsymbol{ k }(\boldsymbol{b}\cdot_{\mathbb{E}^N} \boldsymbol{v})\), where \(\boldsymbol{k},\ \boldsymbol{b}\in \mathbb{E}^N\), Yes
Tensors in this course
Definition We will call all linear operators in which $M=N$ as tensors.