Linear Algebra with functions Basically my question is - How to check for linear independence between functions ?!
Let the group $\mathcal{F}(\mathbb{R},\mathbb{R})$
  Be a group of real valued fnctions.
i.e $\mathcal{F}(\mathbb{R},\mathbb{R})=\left\{ f:\mathbb{R}\rightarrow\mathbb{R}\right\} $
Let 3 functions $f_{1},f_{2},f_{3}$
  be given such that 
$\forall x\in\mathbb{R}\,\,\,f_{1}=e^{x},\,\,f_{2}=e^{2x},\,\,f_{3}=e^{3x}$
$W=sp(f_{1},f_{2},f_{3})$
  what is $dim(W)$
 ?
How to approach this question ? (from a linear algebra perspective)
I know that $\forall x\in\mathbb{R}\,\,\,W=\alpha e^{x}+\beta e^{2x}+\gamma e^{3x}$
And to get the dimension I need to find the base of $W$
so I need to check whether the following holds true :
$\forall x\in\mathbb{R}\,\,\alpha e^{x}+\beta e^{2x}+\gamma e^{3x}=0\,\Leftrightarrow\,\alpha,\beta,\gamma=0$
However when $x=0$
  I get $\alpha+\beta+\gamma=0$
  which leads to infinite amount of solutions.
How to approach this question ?

Hint: Use Wronskian and show that the Wronskian-Determinant does not vansish.
