$P_4(\mathbb{R})$ is the set of polynomials with degree at most $4$ with real coefficients.
$U=\{p \in P_4(\mathbb{R})∣p(3)=0\}$
A basis of $U$ would be $\{(x−3)^4,(x−3)^3,(x−3)^2,(x−3)\}$
But how do we prove, that the set of polynomial is linearly independent?
If we build the linear combination: $a\cdot(x−3)^4 + b\cdot(x−3)^3 + c\cdot(x−3)^2 + d\cdot(x−3) = 0$
Then we see that $a=b=c=d=0$. However, in order to be linearly independent, $a=b=c=d=0$ has to be the only solution for the equation. But this is not the case, since $x=3$ is also a solution $a\cdot(3−3)^4 + b\cdot(3−3)^3 + c\cdot(3−3)^2 + d\cdot(3−3) = 0$.
Could anyone tell me please, where is my mistake?
Thanks