Question

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

${\mathbf{f}}_{\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{5}\mathbf{,}\mathbf{}{\mathbf{f}}_{\mathbf{2}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathbf{cos}}^{\mathbf{2}}\mathbf{x}\mathbf{,}\mathbf{}{\mathbf{f}}_{\mathbf{3}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathbf{sin}}^{\mathbf{2}}\mathbf{x}$

Short answer

The set of function is linearly dependent.

Step-by-step solution

Use wronskian and drive the equation

We have the set of functions

${\mathrm{f}}_1\left(\mathrm{x}\right)=5,{\mathrm{f}}_2\left(\mathrm{x}\right)={\cos }^2\mathrm{x}\mathrm{and}{\mathrm{f}}_3\left(\mathrm{x}\right)={\sin }^2\mathrm{x}$

And we have to determine if this set is linearly independent on the interval or not as the following technique:

First, we have to obtain the wronskian of this set of functions as

$\begin{array}{rcl}\mathrm{W}\left({\mathrm{f}}_1,{\mathrm{f}}_2,{\mathrm{f}}_3\right)& =& \left|\begin{array}{ccc}{\mathrm{f}}_1& {\mathrm{f}}_2& {\mathrm{f}}_3\\ {\mathrm{f}\text{'}}_1& {\mathrm{f}\text{'}}_2& {\mathrm{f}\text{'}}_3\\ {\mathrm{f}\text{'}\text{'}}_1& {\mathrm{f}\text{'}\text{'}}_2& {\mathrm{f}\text{'}\text{'}}_3\end{array}\right|\\ & =& \left|\begin{array}{ccc}5& {\cos }^2\mathrm{x}& {\sin }^2\mathrm{x}\\ 0& -2\sin \mathrm{x}\cos \mathrm{x}& 2\sin \mathrm{x}\cos \mathrm{x}\\ 0& 2{\sin }^2\mathrm{x}-2{\cos }^2\mathrm{x}& 2{\cos }^2\mathrm{x}-2{\sin }^2\mathrm{x}\end{array}\right|\\ & & \end{array}$

Second, we have to find the determinate of this matrix as

$\begin{array}{rcl}\mathrm{W}\left(5,{\cos }^2\mathrm{x},{\sin }^2\mathrm{x}\right)& =& 5\times \left|\begin{array}{cc}-2\mathrm{sinxcosx}& 2\mathrm{sinxcosx}\\ 2{\sin }^2\mathrm{x}-2{\cos }^2\mathrm{x}& 2{\cos }^2\mathrm{x}-2{\sin }^2\mathrm{x}\end{array}\right|-0+0\\ & =& 5\left[(-2\mathrm{sinxcosx})\left(2{\cos }^2\mathrm{x}-2{\sin }^2\mathrm{x}\right)-\left(2\mathrm{sinxcosx}\right)\left(2{\sin }^2\mathrm{x}-2{\cos }^2\mathrm{x}\right)\right]\\ & =& -20{\mathrm{sinxcos}}^3\mathrm{x}+20{\sin }^3\mathrm{xcosx}-20{\sin }^3\mathrm{xcosx}+20{\mathrm{sinxcos}}^3\mathrm{x}\\ & =& \left(-20{\mathrm{sinxcos}}^3\mathrm{x}+20{\mathrm{sinxcos}}^3\mathrm{x}\right)+\left(20{\sin }^3\mathrm{xcosx}-20{\sin }^3\mathrm{xcosx}\right)\\ & =& 0\end{array}$

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

Therefore, the set of functions is linearly dependent.