Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics I / 2 / b / i / Solution

The monomials $1,x,\dots ,x^n$ form a basis of $V_n$: they span every polynomial of degree of a polynomial at most $n$, and equality of polynomials forces equality of every coefficient, proving linear independence. ThereforeThe zero polynomial is included in $V_n$ even though its degree of a polynomial need not be assigned an ordinary nonnegative value.