1. ## linear independence

CAN somebody,please write down the definition for the linear independence of the following functions?

{ $e^x,e^{2x}$}

2. Originally Posted by alexandros
CAN somebody,please write down the definition for the linear independence of the following functions?

{ $e^x,e^{2x}$}
As in any vector space, the set $\{e^x, e^{2x}\}$ is linearly independent iff whenever there are scalars $a,b \in \mathbb{F}$ (where $\mathbb{F}$ is your underlying field) such that $ae^x + be^{2x} =0$ we must have $a=b=0$.

In this particular case putting $x=0$ we have $a=b$ and so $ae^x(1+e^x)=0$ but $0 \neq e^x$ for all $x \in \mathbb{F}$ (assuming $\mathbb{F} = \mathbb{R} or \mathbb{C}$ ) so $a(1+e^x)=0$ for all $x \in \mathbb{F}$ and since $e^x$ is not constant we must have $b=a=0$ and so your set is linearly independent.

3. Your argument is not quite correct. At x=0 we have a+b=0 i.e. a=-b

4. Use the Wronskian and you'll find linear indepencence.

5. Originally Posted by Krizalid
Use the Wronskian and you'll find linear indepencence.
I am sorry i am iterested in the definition and not the proof
thanks

6. Originally Posted by alexandros
I am sorry i am iterested in the definition and not the proof
thanks
Jose27 gave you the definition:

Originally Posted by Jose27
As in any vector space, the set $\{e^x, e^{2x}\}$ is linearly independent iff whenever there are scalars $a,b \in \mathbb{F}$ (where $\mathbb{F}$ is your underlying field) such that $ae^x + be^{2x} =0$ we must have $a=b=0$.