From a bank of past prelim exams:
Let $V$ be the real vector space of real polynomials of degree at most $5$, and define $T: V \to \Bbb{R}^3$ by $T(f) = (f(0), f(1), f(2))$. Find the dimensions of the kernel and image of $T$, and find a basis for each.
I've attempted a solution, working with the standard basis of $V$ and concluded that the kernel is the set of polynomials divisible by $x(x-1)(x-2)$. Thus, I concluded that the dimension of the kernel is $3$, and the dimension of the image is $6-3=3$ by the rank-nullity theorem.
I'm just unsure of how to write the basis of the kernel and image. Any help would be greatly appreciated.