Question

Are the following linear vector functions? Prove your conclusions using (7.2).

4.$\mathit{F}\mathbf{\left(}\mathit{r}\mathbf{\right)}\mathbf{=}\mathit{r}\mathbf{+}\mathit{A}$,whereAis a given vector.

Short answer

The given function$F\left(r\right)=r+A$ is not linear.

Step-by-step solution

Definition of Linear Vector Function:

A vector-valued function, generally termed as a vector function, is a mathematical function with one or more variables that has a range of multidimensional or infinite-dimensional vectors.

A function of a vector, say f(r), is called linear if,$\mathit{f}\mathbf{(}{\mathit{r}}_{\mathbf{1}}\mathbf{+}{\mathit{r}}_{\mathbf{2}}\mathbf{)}\mathbf{=}\mathit{f}\mathbf{\left(}{\mathit{r}}_{\mathbf{1}}\mathbf{\right)}\mathbf{+}\mathit{f}\mathbf{\left(}{\mathit{r}}_{\mathbf{2}}\mathbf{\right)}$, and$\mathit{f}\mathbf{\left(}\mathit{a}\mathit{r}\mathbf{\right)}\mathbf{=}\mathit{a}\mathit{f}\mathbf{\left(}\mathit{r}\mathbf{\right)}$, whereis a scalar.

For example, $F\left(r\right)=br$ (where is a scalar) is a linear vector function of .

Given parameters:

The given vector function is,

$F\left(r\right)=r+A$

Check to see whether the given vector function is linear.

Find out if the given vector function is linear:

Verify that the given vector function is linear with the help of two conditions which are defined as follow.

$$\begin{gathered} F(r_1+r_2)=F\left(r_1\right)+F\left(r_2\right).....\left(1\right) \\ F\left(ar\right)=aF\left(r\right).....\left(2\right) \end{gathered}$$

The first condition's LHS.

$F(r_1+r_2)=r_1+r_2+A$

The first condition's RHS.

$F\left(r_1\right)+F\left(r_2\right)=r_1+A+r_2+A$

The function is not linear because the RHS and LHS are not the same. The second condition is similarly not fulfilled.

$$\begin{gathered} F\left(ar\right)=ar+A\ne aF\left(r\right) \\ =a\left(r+A\right) \end{gathered}$$

Hence, the given function is not linear