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

#### karush

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

View attachment 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

##### Indicium Physicus
Staff member
$$=e^x(-\cos(x)-\cos(x))=-2e^x\cos(x)\dots$$

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

#### skeeter

##### Well-known member
MHB Math Helper
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*}
$$\displaystyle w(e^x,\cos{x}) = \begin{vmatrix} e^x & \cos{x}\\ e^x & -\sin{x} \end{vmatrix} = -e^x(\sin{x}+\cos{x})$$

#### karush

##### Well-known member
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

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