Question

Let $\mathcal{D}^{(1)}$ denote the set of all first order linear differential operators $L=p(x) D+q(x)$ where $p, q$ are polynomials. (a) Prove that $\mathcal{D}^{(1)}$ is a vector space. Is it finite-dimensional or infinite-dimensional? (b) Prove that the commutator (7.17) of $L, M \in \mathcal{D}^{(1)}$ is a first order differential operator $[L, M] \in \mathcal{D}^{(1)}$ by writing out an explicit formula. (c) Verify the Jacobi identity (7.18) for the first order operators $L=D, M=x D+1$, and $N=x^2 D+2 x$.

   Let $\mathcal{D}^{(1)}$ denote the set of all first order linear differential operators $L=p(x) D+q(x)$ where $p, q$ are polynomials. (a) Prove that $\mathcal{D}^{(1)}$ is a vector space. Is it finite-dimensional or infinite-dimensional? (b) Prove that the commutator (7.17) of $L, M \in \mathcal{D}^{(1)}$ is a first order differential operator $[L, M] \in \mathcal{D}^{(1)}$ by writing out an explicit formula. (c) Verify the Jacobi identity (7.18) for the first order operators $L=D, M=x D+1$, and $N=x^2 D+2 x$.

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 7, Problem 49

Instant Answer

Step 1

- Closure under addition: Let $L = p(x)D + q(x)$ and $M = r(x)D + s(x)$ be elements of $\mathcal{D}^{(1)}$. Then their sum is $L + M = (p(x) + r(x))D + (q(x) + s(x))$. Since the sum of polynomials is a polynomial, $p(x) + r(x)$ and $q(x) + s(x)$ are polynomials. Show more…

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Let $\mathcal{D}^{(1)}$ denote the set of all first order linear differential operators $L=p(x) D+q(x)$ where $p, q$ are polynomials. (a) Prove that $\mathcal{D}^{(1)}$ is a vector space. Is it finite-dimensional or infinite-dimensional? (b) Prove that the commutator (7.17) of $L, M \in \mathcal{D}^{(1)}$ is a first order differential operator $[L, M] \in \mathcal{D}^{(1)}$ by writing out an explicit formula. (c) Verify the Jacobi identity (7.18) for the first order operators $L=D, M=x D+1$, and $N=x^2 D+2 x$.

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