Which, if any, of the following polynomials are in Range(t)?

Question

Let T: P^2 ----> P^2 be a linear transformation defined by T(p(x)) = xp'(x)

(i) 2

(ii) x^2

(iii)1-x

I was hoping someone would show me how to find the range of one of them so I know how to do the same for the other two. Thank you for the help!

Answer 1

In general, $$T(ax^2+bx+c)=x(2ax+b)=(2a)x^2+bx+0$$ So the range is all polynomials of the form $$ax^2+bx$$ In other words, all polynomials of second degree or lower without a constant term. Only (ii) is in the range.

Alternatively, let $\{1,x,x^2\}$ be a basis of $\mathbb{P}_2$. Then the matrix for $T$ is $$A=\begin{bmatrix}0&0&0\\0&1&0\\0&0&2\end{bmatrix}$$

Therefore the range is given by $$\text{span}\left(\begin{bmatrix}0\\1\\0\end{bmatrix}\begin{bmatrix}0\\0\\2\end{bmatrix}\right)=\text{span}\left(\begin{bmatrix}0\\1\\0\end{bmatrix}\begin{bmatrix}0\\0\\1\end{bmatrix}\right)$$ which is $$\text{span}(x,x^2)=\{ax^2+bx:a,b\in\mathbb{R}\}$$

Answer 2

If $p=2\in P^2$ then $T(p)=x\cdot 0=0$