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\begin{document}

{\bf Question}

Determine if the functions in the following sets are linearly
independent.
\begin{enumerate}

\item The set $1$, $x$, $x^2$.

\item The set $\cos x$, $\sin x \qquad (*)$

\end{enumerate}


\vspace{0.25in}

{\bf Answer}
\begin{enumerate}

\item Consider the Wronskian for the functions $1$, $x$, $x^2$. $$
W(x)=\left|\begin{array}{ccc}
             1 &amp; x &amp; x^2\\
         0 &amp; 1 &amp; 2x\\
             0 &amp; 0 &amp; 2
    \end{array}\right| = 2
$$ Hence since $W\not\equiv 0$ the functions are linearly
independent.

\item
Consider the Wronskian for the functions $\sin x$, $\cos x$. $$
W(x)=\left|\begin{array}{cc}
             \sin x &amp; \cos x \\
         \cos x &amp; -\sin x
    \end{array}\right| = -\sin^2x - \cos^2x = -1
$$ Hence since $W\not\equiv 0$ the functions are linearly
independent.

\end{enumerate}


\end{document}
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