Let $D : \Bbb{R}[x]^{\le n}\to\Bbb{R}[x]^{\le n}$ be the differentiation map.
Fix a number $a \neq 0$ and let $T : \Bbb{R}[x]^{\le > n}\to\Bbb{R}[x]^{\le n}$ be the map $D + Z_{a}$ (that is, $Tp = > \frac{dp}{dx} + ap$).
Show that T maps the basis of monomials to a set of n + 1 polynomials of distinct degrees.
Is the basis of monomials {$1, x, x^2, ... , x^n$}? If so, applying T will give me {$a, 1 + ax, 2x + 2x^2, ... , nx^{(n-1)} + ax^n$}, which is a set of n polynomials with distinct degrees and not n+1. What am I getting wrong?