Question

In Problems 15–22 determine whether the given set of functions is

linearly independent on the interval$\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$$\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$.

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

Short answer

The solution for the given set of function is linear independent.

Step-by-step solution

Determine the set of function by using Wronskian

We have the set of the function

${\mathrm{f}}_1\left(\mathrm{x}\right)=1+\mathrm{x},{\mathrm{f}}_2\left(\mathrm{x}\right)=\mathrm{x},{\mathrm{f}}_3\left(\mathrm{x}\right)={\mathrm{x}}^2$

Determine the set of function is linearly independent on the interval$(-\infty ,\infty )$or not as the following technique:

Obtaining the Wronskian of the set of function as

$\begin{array}{rcl}\mathrm{W}\left({\mathrm{f}}_1,{\mathrm{f}}_2,{\mathrm{f}}_3\right)& =& \left|\begin{array}{ccc}{\mathrm{f}}_1& {\mathrm{f}}_2& {\mathrm{f}}_3\\ {\mathrm{f}\text{'}}_1& {\mathrm{f}\text{'}}_2& {\mathrm{f}\text{'}}_3\\ {\mathrm{f}}_1\text{'}\text{'}& {\mathrm{f}}_2\text{'}\text{'}& {\mathrm{f}}_3\text{'}\text{'}\end{array}\right|\\ & =& \left|\begin{array}{ccc}1+\mathrm{x}& \mathrm{x}& {\mathrm{x}}^2\\ 1& 1& 2\mathrm{x}\\ 0& 0& 2\end{array}\right|\end{array}$

Determine the matrix as,

$\begin{array}{rcl}\mathrm{W}\left({\mathrm{f}}_1\left({\mathrm{f}}_1,{\mathrm{f}}_2,{\mathrm{f}}_3\right)\right.& =& (1+\mathrm{x})\times \left|\begin{array}{cc}1& 2\mathrm{x}\\ 0& 2\end{array}\right|-\mathrm{x}\times \left|\begin{array}{cc}1& 2\mathrm{x}\\ 0& 2\end{array}\right|+{\mathrm{x}}^2\times \left|\begin{array}{cc}1& 1\\ 0& 0\end{array}\right|\\ & =& (1+\mathrm{x})\left[\right(1\left)\right(2)-(0\left)\right(2\mathrm{x}\left)\right]-\mathrm{x}\left[\right(1\left)\right(2)-(0\left)\right(2\mathrm{x}\left)\right]+{\mathrm{x}}^2\left[\right(1\left)\right(0)-(0\left)\right(1\left)\right]\\ & =& 2(1+\mathrm{x})-2\left(\mathrm{x}\right)+0\\ & =& 2+2\mathrm{x}-2\mathrm{x}\\ & =& 2\\ & & \end{array}$

Since the determinate of the Wronskian of the given set of function is not equal zero, Then this set of function is linearly independent.