A linear transformation from a vector space of polynomials

#linear_algebra

Let $V$ denote the real vector space of polynomials in $x$ of degree at most 3. Let $B = {1, x, x^{2}, x^{3}}$ be a basis for $V$ and $T : V \to V$ be the function defined by $T (f (x)) = f (x) + f^{'} (x)$ .

Prove that $T$ is a linear transformation.
Find $[T]_{B}$ , the matrix representation for $T$ in terms of the basis $B$ .
Is $T$ diagonalizable? If yes, find a matrix $A$ so that $A [T]_{B} A^{- 1}$ is diagonal, otherwise explain why $T$ is not diagonalizable.

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Let $V$ denote the real vector space of polynomials in $x$ of degree at most 3. Let $\mathcal{B}=\{1, x, x^2, x^3\}$ be a basis for $V$ and $T:V\to V$ be the function defined by $T(f(x))=f(x)+f'(x)$.
\begin{enumerate}[label=(\alph*)]
	\item Prove that $T$ is a linear transformation.
	\item Find $[T]_{\mathcal{B}}$, the matrix representation for $T$ in terms of the basis $\mathcal{B}$.
	\item Is $T$ diagonalizable? If yes, find a matrix $A$ so that $A[T]_{\mathcal{B}}A^{-1}$ is diagonal, otherwise explain why $T$ is not diagonalizable.
\end{enumerate}