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Discuss the dependence of the set $\left\{e^{x}, e^{2 x}, e^{-2 x}, \sin h 2 x\right\}$

$\sinh (x)=\frac{e^{x}-e^{-x}}{2}$

$\sinh (2 x)=\frac{e^{2 x}-e^{-2 x}}{2}$

$\sinh (2 x)=\frac{e^{2 x}}{2}-\frac{e^{-2 x}}{2}=\frac{1}{2} e^{2 x}-\frac{1}{2} e^{-2 x}+0 e^{x}$

Linearly dependent

Let $f(t)=t^{2}|t|$ and $g(t)=t^{3}$
Show that $w(f, g)=0$ where $t \in(-1,1)$

$t=0$

$f(t)=0 \quad, \quad g(t)=0, \quad w(f, g)=0$

$-1<t<0$

$f=-t^{3}, g(t)=t^{3}, \quad f=-g \rightarrow{f, g}$ is $L . D$
$w(f, g)=0$

$0<t<1$

$f=t^{3}, g=t^{3} \rightarrow f=g \rightarrow\left\{f,{g}\right\} \text { is L.D }$

$w(f, g)=0$

$w(f, g)=0$

Determine whether the functions are linearly independent or
linearly dependent, If they are linearly dependent, find a linear
relation among them.
$\left\{1-t+2 t^{2},-1+t^{2},-2-t+5 t^{2}\right\}$

$c_{1} f_{1}+c_{2} f_{2}+c_{3} f_{3}=0$

$c_{1}\left(1-t+2 t^{2}\right)+c_{2}\left(-1+t^{2}\right)+c_{3}\left(-2-t+5 t^{2}\right)=0$

$c_{1}-c_{1} t+2 c_{1} t^{2}+-c_{2}+c_{2} t^{2}-2 c_{3}-c_{3} t+5 c_{3} t^{2}=0$

$t^{2}\left(2 c_{1}+c_{2}+5 c_{3}\right)+t\left(-c_{1}-c_{3}\right)+\left(c_{1}-c_{2}-2 c_{3}\right)=0$

$2 c_{1}+c_2+5 c_3=0$
$-c_{1}-c_{3}=0$
$c_{1}-c_{2}-2 c_{3}=0$

$[A]=\left[\begin{array}{ccc}{2} & {1} & {5} \\ {-1} & {0} & {-1} \\ {1} & {-1} & {-2}\end{array}\right]$

$C_{1}=-k$
$C_{2}=-3 k$
$C_{3}=k$

The set is linearly dependent

$k=1, c_{1}=-1, c_{2}=-3, c_{3}=1$

$-f_{1}-3 f_{2}+f_{3}=0$