CAN somebody,please write down the definition for the linear independence of the following functions?
{$\displaystyle e^x,e^{2x}$}
As in any vector space, the set $\displaystyle \{e^x, e^{2x}\}$ is linearly independent iff whenever there are scalars $\displaystyle a,b \in \mathbb{F}$ (where $\displaystyle \mathbb{F}$ is your underlying field) such that $\displaystyle ae^x + be^{2x} =0$ we must have $\displaystyle a=b=0$.
In this particular case putting $\displaystyle x=0$ we have $\displaystyle a=b$ and so $\displaystyle ae^x(1+e^x)=0$ but $\displaystyle 0 \neq e^x$ for all $\displaystyle x \in \mathbb{F}$ (assuming $\displaystyle \mathbb{F} = \mathbb{R} or \mathbb{C}$ ) so $\displaystyle a(1+e^x)=0$ for all $\displaystyle x \in \mathbb{F}$ and since $\displaystyle e^x$ is not constant we must have $\displaystyle b=a=0$ and so your set is linearly independent.