# Determine whether the given set of functions is linearly dependent or linearly i

Determine whether the given set of functions is linearly dependent or linearly independent on the interval $\left(-\mathrm{\infty },\mathrm{\infty }\right)$
${f}_{1}\left(x\right)=1+x,$

${f}_{2}\left(x\right)=x,$

${f}_{3}\left(x\right)={x}^{2}$

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From Theorem we know that the function ${f}_{1},{f}_{2}$, and ${f}_{3}$ are linearly independent if and only if $W\left({f}_{1},{f}_{2},{f}_{3}\right)\ne 0$
Where $W\left({f}_{1},{f}_{2},{f}_{3}\right)=|\begin{array}{ccc}{f}_{1}& {f}_{2}& {f}_{3}\\ {f}_{1}^{\prime }& {f}_{2}^{\prime }& {f}_{3}^{\prime }\\ {f}_{1}^{″}& {f}_{2}^{″}& {f}_{3}^{″}\end{array}|$

$\therefore W\left({f}_{1},{f}_{2},{f}_{3}\right)=|\begin{array}{ccc}1+x& x& {x}^{2}\\ 1& 1& 2x\\ 0& 0& 2\end{array}|$
Using the properties of determinants we have
$W\left({f}_{1},{f}_{2},{f}_{3}\right)=\left(1+x\right)|\begin{array}{cc}1& 2x\\ 0& 2\end{array}|-x|\begin{array}{cc}1& 2x\\ 0& 2\end{array}|+{x}^{2}|\begin{array}{cc}1& 1\\ 0& 0\end{array}|$
$W\left({f}_{1},{f}_{2},{f}_{3}\right)=\left(1+x\right)\left(2-0\right)-x\left(2-0\right)+{x}^{2}\left(0-0\right)$
$W\left({f}_{1},{f}_{2},{f}_{3}\right)=\left(1+x\right)\left(2\right)-x\left(2\right)+{x}^{2}\left(0\right)$
$W\left({f}_{1},{f}_{2},{f}_{3}\right)=2+2x-2x+0\to W\left({f}_{1},{f}_{2},{f}_{3}\right)=2$
$W\left({f}_{1},{f}_{2},{f}_{3}\right)\ne 0$
Thus, the given function are linearly independent
Result: The given functions are linearly independent As their Wronskian $W\left({f}_{1},{f}_{2},{f}_{3}\right)=2\ne 0$

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