Linear Algebra - Determinant of linear transformation

Question

So I'm working through sample questions and this came up. Any help would be greatly appreciated.

Question

Let $V$ be the vector space of all complex-valued polynomials $p(x)$ of degree at most $42$ and consider the linear transformation $L\colon V\to V$ given by:

$L(p(x)) = -p^{\prime\prime}(x) + 22p(x)$, where $p^{\prime\prime}$ is the second derivative of $p$.

(a)State the definition of $\det(L)$

(b)Find $\det(L)$. Justify your answer!

Progress

(a) I used Leibniz' definition where $$\det(L) = \sum_{\sigma\in S_n}sgn(\sigma)\prod^n_{i=1}a_{i,\sigma(i)}$$ for some $a\in L$.

(b) I know how to get the determinant for a matrix but I don't understand it here. Thanks in advance for anyone who can help.

Answer 1

Hints:

$$\begin{align}&L(1)=22\\ &L(x)=22x\\ &L(x^2)=2+22x^2\\ &\ldots\;\ldots\;\ldots\\ &L(x^k)=k(k-1)x^{k-2}+22x^k\\ &\ldots\;\ldots\;\ldots\\ &L(x^{42})=42\cdot41x^{40}+22x^42\end{align}\;\;\;\implies\;\;[L]^t=\begin{pmatrix}22&0&0&\ldots&0\\ 0&22&0&\ldots&0\\ \ldots&\ldots&\ldots&\ldots&\ldots\\ 0&\ldots&42\cdot41&0&22\end{pmatrix}$$

Since $\;\det A=\det A^t\;$ for any matrix, the above is enough to know what you need.