Consider the function $d/dx$ from $P_n$ (the real vector space of degree $\leq n$ polynomials in one variable $x$) to $P_{n-1}$.
a) Prove that $d/dx$ is a linear transformation.
b) Write the associated matrix of the linear transformation with respect to the standard basis $(\{1,x,x^2,\dots,\})$ for $P_n$ and $P_{n-1}$.
c) What is the kernel of $d/dx$? What is range of $d/dx$?
I proved that $d/dx$ is a linear transformation by showing that it holds under additivity and homogeneity. However, I'm stuck on b). I tried to align the standard basis with its derivatives however I am confused as to what to do for the derivatives of $1$ and $x$.
I know that the kernel of $d/dx$ is all constants and the range is going to be $P_{n-1}$.