Define the mapping $T$to be the one that maps a polynomial $f(x) \in V$ to the vector $(f(0), f(1),f(2),f(3),f(4),f(5))^t$, where $V$ is the vector space of all real polynomials of degree 5 or less.
(I've been told to think about $f$ being in the null space.)
I'm thinking that $T$ has rank 6 and dim(null space) = 0. But I'm not real sure as to how to make it precise.
2nd part: Is it the case that for any sequence of 6 numbers $a_0, \dots a_5$ we have that there exists an $f(x) \in V$ such that $f(0) = a_0 ,\dots, f(1) = a_5$. (I know this is saying something about polynomial interpolation.)
Suppose $[f(0)\ f(1)\ f(2)\ f(3)\ f(4)\ f(5)]^t=0$; then $f$ has (at least) six roots; but $f$ has degree at most $5$, so…
Now rank-nullity tells you what the rank is.
(Hint: your conjecture is right.)