determining the matrix of a linear transformation (first order derivative) with respect to a basis

Question

Let V be the vector space of all polynomials of degree ≤ $k$.
Let $D:V→V$ be the linear transformation given by $p(x) → p′(x)$ (the derivative).
Determine the matrix of $D$ with respect to the basis $1, x, . . . , x^k$ and determine the rank and nullity of $D$.

I'm unsure of how to approach this problem. Should I construct a jacobian matrix? If the basis is $1,x,...,x^k$ then all components of the transformation matrix $D$ should be able to be written as $d$1,$d$x,...,$d$$x^k$ but how that would fit with the derivatives is unclear.

Answer 1

Your basis is $\{1,x,x^2,\ldots,x^k\}$. Furthermore,

• $D(1)=0$;
• $D(x)=1$;
• $D(x^2)=2x$;
• $\cdots$
• $D(x^k)=kx^{k-1}$.

Therefore your matrix is$$\begin{bmatrix}0&1&0&0&\ldots&0\\0&0&2&0&\ldots&0\\0&0&0&3&\ldots&0\\\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\0&0&0&0&\cdots&k\\0&0&0&0&\ldots&0\end{bmatrix}.$$Can you do the rest?