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Linear Algebra

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Matrix of the Derivative Operator

LINALG-QIE7KE

Let $V$ be the vector space of polynomials of degree at most $3$ in the variable $x$, and let $T:V\to V$ be the linear transformation:

$$T(f)=\cfrac{df}{dx}$$

Find the matrix of $T$ with respect to the basis:

$$\mathcal{B}=\{1,x,x^2,x^3\}$$

...of $V$.

A

$\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{pmatrix}$

B

$\begin{pmatrix}0&0&0&0\\\ 0&1&0&0\\\ 0&0&2&0\\\ 0&0&3&0\end{pmatrix}$

C

$\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&2&0\\\ 0&0&3&0\end{pmatrix}$

D

$\begin{pmatrix}0&0&0&0\\\ 1&0&0&0\\\ 0&2&0&0\\\ 0&0&3&0\end{pmatrix}$

E

$\begin{pmatrix}0&1&0&0\\\ 0&0&2&0\\\ 0&0&0&3\\\ 0&0&0&0\end{pmatrix}$