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.