$T(af+bg)=0$ for all scalars $a,b$ and $f\neq kg$ for all scalar $k$

Question

I saw the following statement somewhere:

Let $T:P_{n}\longrightarrow X$ be linear and onto, let $\dim X=2$ and $n\geq 3.$ Prove that there exist $f,g\in P_{n}$ such that $T(af+bg)=0$ for all real numbers $a,b$ and $f\neq kg$ for all real $k.$

In my opinion, if we choose first, $b=a\neq 0$, secondly $b=-a=\neq 0$ then $Tf$ must be $0$ using the same argument $Tg=0.$

Can someone confirm this? Prove or disprove!

Thank you

Answer 1

Since $\operatorname{dim}P_n\ge4$, we have $\operatorname{nullity}T\ge2$ by the Rank-nullity theorem. $\therefore \exists f,g\in P_n$, linearly independent, with $T(f)=T(g)=0$.