TITLE: How to show that the set of functions $1,x,x^2,x^3...$ is linearly independent?
QUESTION [5 upvotes]: Show that the set of functions $1,x,x^2,x^3...x^n...$ is linearly independent on any interval $[a,b]$.
If $$c_1+xc_2+x^2c_3+x^3c_4...=0$$ we should show $$c_i=0,\quad i=1,2, \ldots$$
how could I start?
My second question: is it linearly independent on $C[0,1]$?

REPLY [5 votes]: Note that if coefficients $a_{0},...,a_{n}$ exist s.t
$$
p(x)=a_{0}+a_{1}x+...+a_{n}x^{n}=0;\,\forall x\in[a,b]
$$
Then since a polynomial is a continues function it means that 
$$
p(x)=0;\,\forall x\in\mathbb{R}
$$
We now prove by induction that $a_{m}=0$ for all $m$.
Our base case will be $m=n$ and we proceed similarly for $m=n-1$
etc'.
If $a_{n}\neq0$ then 
$$
\lim_{x\to\infty}p(x)=\begin{cases}
\infty & a_{n}>0\\
-\infty & a_{n}<0
\end{cases}
$$
and thus $a_{n}=0$.
When we reach the step $$a_{n}=a_{n-1}=...=a_{1}=0$$ we will be left
with $p(x)=a_{0}$ and since $$a_{0}=p(0)=0$$ we reach the conclusion
$p\equiv0$.
This shows that if we have a linear combination that is zero then
all coefficients are also zero, which shows the independence by definition.