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.

1. $\mathit{f}\left(\text{r}\right)\mathbf{=}\mathbf{\text{A}}\mathbf{·}\mathbf{\text{r}}\mathbf{+}\mathbf{3}$, where A is a give vector.

Short answer

The function f(r) is not linear.

Step-by-step solution

Equations for the linear function

A function of a vector, say , 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 isand it is$f\left(\text{r}\right)=\text{A}·\text{r}+\text{3}$ to be checked whether the function is linear.

The Left-Hand Side (LHS) of the equation $f\left({\text{r}}_{\text{1}}+{\text{r}}_{\text{2}}\right)=f\left({\text{r}}_{\text{1}}\right)+f\left({\text{r}}_{\text{2}}\right)$.

$f\left({\text{r}}_{\text{1}}+{\text{r}}_{\text{2}}\right)=\text{A}·\left({\text{r}}_1+{\text{r}}_{\text{2}}\right)+3$

The Right-Hand Side (RHS) of the equation $f\left({\text{r}}_{\text{1}}+{\text{r}}_{\text{2}}\right)=f\left({\text{r}}_{\text{1}}\right)+f\left({\text{r}}_{\text{2}}\right)$.

$\begin{array}{rcl}f\left({\text{r}}_{\text{1}}\right)+f\left({\text{r}}_{\text{2}}\right)& =& \text{A}·{\text{r}}_1+\text{3}+\text{A}·{\text{r}}_2+\text{3}\\ & =& \text{A}·\left({\text{r}}_1+{\text{r}}_{\text{2}}\right)+\text{6}\end{array}$

As LHS is not equal to RHS, so is does not satisfy the first equation. Therefore, the function is not linear.

Check for the second equation

The Left-Hand Side (LHS) of the equation $f\left(a\text{r}\right)=af\left(\text{r}\right)$.

$f\left(a\text{r}\right)=a\text{A}·\text{r}+3$

The Right-Hand Side (RHS) of the equation $f\left(a\text{r}\right)=af\left(\text{r}\right)$.

$\begin{array}{rcl}af\left(\text{r}\right)& =& a\left(\text{A}·\text{r}+3\right)\\ & =& a\text{A}·\text{r}+3a\end{array}$

Thus, the function does not satisfy both the equations.