A set of functions $\{f_1 , f_2 ,...... , f_n \}$ is called linearly independent on an Interval iff following condition holds.
$\sum_i^n c_i f_i(\alpha) = 0$ for all $\alpha \in I$ iff $c_i=0$ for all $i$.
Can anyone tell me if I have gone wrong anywhere ?
I would alter the definition to the following.
A set of distinct functions $$\{f_1,f_2,......,f_n\}$$ is called linearly independent on an Interval iff following condition holds.
$$∑_{i=1}^n c_i f_i(x)\equiv 0 \iff c_i=0 , 1\le i\le n.$$