Question

In Problems 15–22 determine whether the given set of functions is linearly independent on the interval $\left(-\infty ,\infty \right)$ .

${\mathbf{f}}_{\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}{\mathbf{f}}_{\mathbf{2}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{x}\mathbf{,}{\mathbf{f}}_{\mathbf{3}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathbf{e}}^{\mathbf{x}}$

Short answer

The set of functions is linearly dependent in the interval $\left(-\infty ,\infty \right)$.

Step-by-step solution

Define Wronskian of the function

Consider each of the functions ${\mathrm{f}}_1\left(\mathrm{x}\right),{\mathrm{f}}_2\left(\mathrm{x}\right),...,{\mathrm{f}}_{\mathrm{n}}\left(\mathrm{x}\right)$ possesses at least n - 1 derivatives, then the determinant $\mathrm{W}({\mathrm{f}}_1,{\mathrm{f}}_1,...,{\mathrm{f}}_1)=\left|\begin{array}{cccc}{\mathrm{f}}_1& {\mathrm{f}}_2& ...& {\mathrm{f}}_{\mathrm{n}}\\ {\mathrm{f}\text{'}}_1& {\mathrm{f}\text{'}}_2& ...& \mathrm{f}{\text{'}}_{\mathrm{n}}\\ .& .& .& \mathrm{M}\\ {{\mathrm{f}}^{(\mathrm{n}-1)}}_1& {{\mathrm{f}}^{(\mathrm{n}-1)}}_2& ...& {{\mathrm{f}}^{(\mathrm{n}-1)}}_{\mathrm{n}}\end{array}\right|$ , where the primes denote derivatives is known as the Wronskian of the functions.

Find the Wronskian of the set of functions

The given set of functions is:

$$\begin{gathered} {\mathrm{f}}_1\left(\mathrm{x}\right)=0 \\ {\mathrm{f}}_2\left(\mathrm{x}\right)=\mathrm{x} \\ {\mathrm{f}}_3\left(\mathrm{x}\right)={\mathrm{e}}^{\mathrm{x}} \end{gathered}$$

Using the definition of the Wronskian of the function,

$\begin{array}{rcl}\mathrm{W}\left({\mathrm{f}}_1,{\mathrm{f}}_2,{\mathrm{f}}_3\right)& =& \left|\begin{array}{ccc}{\mathrm{f}}_1& {\mathrm{f}}_2& {\mathrm{f}}_3\\ {\mathrm{f}\text{'}}_1& {\mathrm{f}\text{'}}_2& {\mathrm{f}\text{'}}_3\\ {\mathrm{f}}_1\text{'}\text{'}& {\mathrm{f}}_2\text{'}\text{'}& {\mathrm{f}}_3\text{'}\text{'}\end{array}\right|\\ & =& \left|\begin{array}{ccc}0& \mathrm{x}& {\mathrm{e}}^{\mathrm{x}}\\ 0& 1& {\mathrm{e}}^{\mathrm{x}}\\ 0& 0& {\mathrm{e}}^{\mathrm{x}}\end{array}\right|\end{array}$

Find the determinant of the matrix

Determining the determinant of the matrix described by the Wronskian,

$\begin{array}{rcl}\mathrm{W}\left({\mathrm{f}}_1,{\mathrm{f}}_2,{\mathrm{f}}_3\right)& =& 0\times \left|\begin{array}{cc}1& {\mathrm{e}}^{\mathrm{x}}\\ 0& {\mathrm{e}}^{\mathrm{x}}\end{array}\right|-\mathrm{x}\times \left|\begin{array}{cc}0& {\mathrm{e}}^{\mathrm{x}}\\ 0& {\mathrm{e}}^{\mathrm{x}}\end{array}\right|+{\mathrm{e}}^{\mathrm{x}}\times \left|\begin{array}{cc}0& 1\\ 0& 0\end{array}\right|\\ & =& 0+0+0\\ & =& 0\end{array}$

Since the value of the determinant is zero and the Wronskian of the set of functions is zero,

Hence, the set of functions is linearly dependent.