Question

Question: Are the following linear functions? Prove your conclusions by showing that f(r)satisfies both of the equations (7.1) or that it does not satisfy at least one of them.

2.$\mathit{f}\left(\text{r}\right)\mathbf{=}\mathbf{\text{A}}\mathbf{·}\left(\text{r}-\text{k}z\right)$.

Short answer

The function f(r) is linear.

Step-by-step solution

Equations for the linear function

A function of a vector, say f(r), is called linear if, $f\left(r_1+r_2\right)=f\left(r_1\right)+f\left(r_2\right)$, and $f\left(ar\right)=af\left(r\right)$exists, where a is a scalar.

Check for the first equation

The given function is$f\left(\text{r}\right)=\text{A}·\left(\text{r}-\text{k}z\right)$ and it is to be checked whether the function is linear.

The given function is re-written as$f\left(\text{r}\right)=\text{A}·\left(\text{r}-\text{k}\left(\text{k}·\text{r}\right)\right)$.

Find the first equation of linearity.

$\begin{array}{rcl}f\left({\text{r}}_{\text{1}}+{\text{r}}_{\text{2}}\right)& =& \text{A}·\left(\left({\text{r}}_1+{\text{r}}_{\text{2}}\right)-\text{k}\left(\text{k}·\left({\text{r}}_1+{\text{r}}_{\text{2}}\right)\right)\right)\\ & =& \text{A}·\left({\text{r}}_1-\text{k}\left(\text{k}·{\text{r}}_1\right)\right)+\text{A}·\left({\text{r}}_2-\text{k}\left(\text{k}·{\text{r}}_2\right)\right)\\ & =& f\left({\text{r}}_{\text{1}}\right)+f\left({\text{r}}_2\right)\end{array}$

The function satisfies the first equation.

Check for the second equation

The second equation of linearity.

$\begin{array}{rcl}f\left(a\text{r}\right)& =& \text{A}·\left(a\text{r}-\text{k}\left(\text{k}·a\text{r}\right)\right)\\ & =& a\text{A}·\left(\text{r}-\text{k}\left(\text{k}·\text{r}\right)\right)\\ & =& af\left(\text{r}\right)\end{array}$

Thus, the function satisfies the second equation.

Since, the function satisfies both the equation hence, the function is linear.