Question

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

${\mathbf{\text{F}}}_{\mathbf{\text{1}}}\left(\text{x}\right)\mathbf{\text{= x,}}\mathbf{}{\mathbf{\text{F}}}_{\mathbf{\text{2}}}\left(\text{x}\right){\mathbf{\text{= x - 1,F}}}_{\mathbf{\text{3}}}\left(\text{x}\right)\mathbf{\text{= x - 3}}$

Short answer

Given set of functions is linearly dependent

Step-by-step solution

Use wronskian and drive the equation

${\text{F}}_{\text{1}}\left(\text{x}\right){\text{= x,F}}_{\text{2}}\left(\text{x}\right){\text{= x - 1,F}}_{\text{3}}\left(\text{x}\right)\text{= x - 3}$

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

$\begin{array}{rcl}\mathrm{W}\left({\mathrm{f}}_1,{\mathrm{f}}_2,{\mathrm{f}}_3\right)& =& \left|\begin{array}{ccc}{\mathrm{f}}_1& {\mathrm{f}}_2& {\mathrm{f}}_3\\ {\mathrm{f}\text{'}}_1& {\mathrm{f}\text{'}}_2& {\mathrm{f}\text{'}}_3\\ {\mathrm{f}}_1\text{'}\text{'}& {\mathrm{f}}_2\text{'}\text{'}& {\mathrm{f}}_3\text{'}\text{'}\end{array}\right|\\ & =& \left|\begin{array}{ccc}x& \mathrm{x}-1& x-3\\ 1& 1& 1\\ 0& 0& 0\end{array}\right|\end{array}$

Second we have to find the determinate of this matrix as

$\begin{array}{rcl}\mathrm{W}\left({\mathrm{f}}_1,{\mathrm{f}}_2,{\mathrm{f}}_3\right)& =& \mathrm{x}\left(0\right)-(\mathrm{x}-1)\left(0\right)+(\mathrm{x}-3)\left(0\right)\\ & =& 0\end{array}$

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.