For two functions to be linearly independent we should have this statement:
The functions $f_1, \ldots , f_k$ are said to be linearly independent on $I$, if they are not dependent. Equivalently, whenever $\alpha_1, \ldots , \alpha_k \in \mathbb R$ are such that $$\alpha_1 f_1(x) + \alpha_2 f_2(x) + \cdots + \alpha_k f_k(x) = 0, \forall x \in I,$$ we have $\alpha_1 = \alpha_2 = \cdots = \alpha_k = 0$.
My question: does the above definition imply that $\alpha_1 f_1(x) + \alpha_2 f_2(x) + \cdots + \alpha_k f_k(x) = f(x)$ in order to be linearly independent?
$\{f_n(x)\}$ is a set of independent functions,
then
$a_1 f_1(x) + a_2f_2(x)+\cdots + a_n f_n(x) = 0 \iff a_1, \cdots, a_n = 0$
What you can say is, $\{f_n(x)\}$ forms a basis for some space, and
$f(x)= a_1 f_1(x) + a_2f_2(x)+\cdots + a_n f_n(x)$ is in the space.