## The Range and Nullspace of the Linear Transformation $T (f) (x) = x f(x)$

## Problem 672

For an integer $n > 0$, let $\mathrm{P}_n$ be the vector space of polynomials of degree at most $n$. The set $B = \{ 1 , x , x^2 , \cdots , x^n \}$ is a basis of $\mathrm{P}_n$, called the standard basis.

Let $T : \mathrm{P}_n \rightarrow \mathrm{P}_{n+1}$ be the map defined by, for $f \in \mathrm{P}_n$,

\[T (f) (x) = x f(x).\]

Prove that $T$ is a linear transformation, and find its range and nullspace.

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