Let $\mathbb R_n[X]$ be the vector space of polynomials of degree at most $n$. Let $u$ be the endomorphism $$u(P)=(X^2-1)P''-2XP'$$
I want to determine the determinant of $u$. So I proceed by computing the images of the basis $(1,X,\cdots ,X^n)$. But since $u(1)=0$ it is clear that matrix of $u$ will have a zero column so the determinant is zero. Is this correct? Is it always true that any endomorphism having one element of the basis in its kernel has determinant zero?