How to prove that ${1,x,x^2}$ is a basis of a real polynomial functions space. Let $V$ be the real vector space of all polynomial functions from $\mathbb{R}$ to $\mathbb{R}$ at most second degree. That is, the space of all functions with form $f(x)=c_0+c_1x+c_2x^2$ with $c_i\in\mathbb{R}$

I need to prove that $\{1,x,x^2\}$ is a basis of $V$. But for that, first I need to prove the linear independence of those vectors, right? Well, here is where I get stuck. The only way I can think to do it, is defining an isomorphism $ T:V \to \mathbb{R^3} $ where each coeficient $ T(f)=(c_1,c_2,c_3) $ and proof the linear independece on $\mathbb{R^3}$.

Despite it work, probably it would be better if the proof is made without using transformations. Any suggestions?
hint: Suppose that $a_1\cdot 1 + a_2\cdot x + a_3\cdot x^2 = 0, \forall x \in \mathbb{R}$, then substitute $x = 0,1,2$ into the equation and solve for $a_1,a_2,a_3$. What do you get?
First, $V$ is the space of polynomials of degree $\le 2,$ not the space of all polynomials. Suppose $c_0+c_1x+ c_2x^2$ is the zero function. Doesn't that imply all $c_j=0$?