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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 are correct on every account except one, which is that you now end up with a set of $n+1$ polynomials. The highest degree is $n$, but you can notice that the lowest degree is $0$. In summary, you have your answer.2012-10-03
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    However I count my bags, I always end up with $4$. Where is the fifth? $0,1,2,3,4$2012-10-06

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