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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?

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You indeed got a set of $n+1$ polynomials. We denote $m_k:=x^k$. Then for $k\geq 1$, $T(m_k)=km_{k-1}+am_k$, which is of degree $k$ and for $k=0$, $m_0=am_0$. So the family $\{m_0,m_1\dots,m_n\}$ is a family of $n+1$ polynomials of distinct degrees.