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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

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

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

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