$\underline{Def}$ (Linear Transformation)
Let $V,W$ be vector spave over $\mathcal{F}$.
A function $T \colon V \rightarrow W$ is called a linear map (transformation) if $\forall x,y \in V,\, \forall c \in \mathcal{F}$,
$(a) T(x+y)=T(x)+T(y)$,
$(b) T(cx)=cT(x)$.
$\underline{Rmk}$
$(1)$ If $T \colon V \rightarrow W$ is linear, then $T(0)=0$.
$\because T(0)=T(0+0)=T(0)+T(0) \Rightarrow T(0)=0$
$(2)$ $T$ is linear iff $\forall x,y \in V,\, \forall c \in \mathcal{F},\, T(cx+y)=cT(x)+T(y)$.
$(3)$ $T \colon V \rightarrow W$ is linear iff $\displaystyle \forall n \in \mathcal{N},\,\forall x_{1}, \cdots, x_{n} \in V,\, \forall a_{1}, \cdots, a_{n} \in \mathcal{F}, T(\sum_{i=1}^{n} a_{i}x_{i})=\sum_{i=1}^{n} a_{i}T(x_{i})$.
In $R^{2}$, some geometric function like rotation, reflection, projection are linear map.
$\underline{Ex}$
Define a map $T \colon M_{m \times n}(\mathbb{F}) \rightarrow M_{n \times m}(\mathbb{F})$ by $T(A)=A^{t},\, \forall A \in M_{m \times n}(\mathbb{F})$. Then $T$ is linear.
$\underline{Notation}$
For each integer $k \geq 0$, let $C^{k}(\mathbb{R})=\{f \colon \mathbb{R} \rightarrow \mathbb{R} \colon\,f, f^{1}, \cdots, f^{(k)}\; exist\;and\;are\;continuous\;\in\;\mathbb{R} \}$.
Then $C^{k}(\mathbb{R})$ is a real vector space $\forall k \geq 0$.
Note $\displaystyle \frac{d}{dx} \colon C^{1}(\mathbb{R}) \rightarrow C(\mathbb{R}),\; \frac{d^{2}}{dx^{2}} \colon C^{2}(\mathbb{R}) \rightarrow C(\mathbb{R}), \cdots, \frac{d^{k}}{dx^{k}} \colon C^{k}(\mathbb{R}) \rightarrow C(\mathbb{R})$ are all linear.
$\underline{Ex}$
Let $a<b$ be any two fixed reals.
Define a map $T \colon C(\mathbb{R}) \rightarrow \mathbb{R}$ by $\displaystyle T(f)=\int_{a}^{b} f(x) dx,\, \forall f \in C(\mathbb{R})$. Then $T$ is linear.
$\underline{Rmk}$ (Two basic linear maps)
Let $V, W$ be vector space over $\mathbb{F}$.
Identity map of $V$: $I_{V} \colon V \rightarrow V$ by $I_{V}(x)=x,\; \forall x \in V$.
Zero map: $T_{0} \colon V \rightarrow W$ by $T_{0}(x)=0,\; \forall x \in V$.
These are linear.