MHB 16.1 Show that e^{2x}, sin(2x) is linearly independent on + infinity -infinity

[karush]

16.1 Show that $e^{2x}$, sin(2x) are linearly independent on $(-\infty,+\infty)$

https://www.physicsforums.com/attachments/9064
that was the example but...

\begin{align*}
w(e^x,\cos x)&=\left|\begin{array}{rr}e^x&\cos{x} \\ e^x&-\cos{x} \\ \end{array}\right|\\
&=??\\
&=??
\end{align*}

 

[Ackbach]

$$=e^x(-\cos(x)-\cos(x))=-2e^x\cos(x)\dots$$

But don't you need to calculate $w\left(e^{2x},\cos(2x)\right)?$ What do you get for that?

 

[skeeter]

but...

\begin{align*}
w(e^x,\cos x)&=\left|\begin{array}{rr}e^x&\cos{x} \\ e^x&-\cos{x} \\ \end{array}\right|\\
&=??\\
&=??
\end{align*}

$$w(e^x,\cos{x}) = \begin{vmatrix}
e^x & \cos{x}\\
e^x & -\sin{x}
\end{vmatrix} = -e^x(\sin{x}+\cos{x})$$

 

[karush]

Sorry everybody I think this thread went off the rails
my 2nd post was way off
so Ill post another new one of a similar problem