I was thinking about how to determine if two functions $f,g$ defined on $\mathbb{R}$ are linearly independent. For example, $f(x)=x^3$ and $g(x)=x^5$ then there is no way that these two functions are linearly dependent. However, for much more complicated functions, the linear independence cannot be easily determined (at least in my opinion). Therefore, I need to express these functions as linear combinations of their basis.
My primary idea is to expand a function by its Taylor series, and thus $f(x)$ would be an element of degree $\infty$ in $\mathbb{R}[[x]]$. The function could be written as a vector $$\left\langle f(0),f'(0),\frac{f''(0)}{2!},\dots\right\rangle$$ Therefore, if we have two functions $f,g$, then the linear independence can be quickly determined by checking if there exists a pair $(c_1,c_2)$ s.t. $\forall n$, $$c_1{f^{(n)}(0)}+c_2{g^{(n)}(0)}=0$$ Here are some doubts of mine:
(1) Can we determine an explicit basis for $\mathbb{R}[[x]]$?
(2) Can we check the linear independence of two $\infty$-dimensional vectors?
(3) Is my idea correct?
Thanks in advance. Please don't be too harsh if this is a stupid idea.