4 Chapter 1: Linear systems and their description

Let $𝐂$ be the field of complex numbers, let $V$ and $W$ be vector spaces over $𝐂$, so $\lambda f+\mu g\in V$ for all $f,g\in V$ and $\lambda ,\mu \in 𝐂$. Time is $t>0$. A map $L:V\to W$ is called linear if

$$L(\lambda f+\mu g)=\lambda Lf+\mu Lg.$$

The following give the basic examples with

$$V=W=\{\text{continuously differentiable functions}\mathit{\hspace{1em}}f:[0,\mathrm{\infty })\to 𝐂\}.$$

i) Differentiation $Lf=\frac{df}{dt};$ symbolised by $[d/dt]$;

ii) Integration $Lf(x)={\int }_0^{x}f(t)𝑑t,$ symbolised as $[\int ]$;

iii) an amplifier is multiplication by $a\in 𝐂,$ symbolised as $[a]$;

iv) Multiplication by $h\in V$, $Lf(t)=h(t)f(t)$ symbolised as $[h]$;

v) Evaluation at $t_0$, $f\mapsto f(t_0)$ symbolised as $[{\delta }_{t_0}]$.