Referring to Example 24.3, show that YX - XY = 1. 

Data from in 24.3 Example 

Let F be a field of characteristic zero, and let (F[x], +) be the additive group of the ring F[x] of polynomials with coefficients in F. For this example, let us denote this additive group by F[x], to simplify this notation. We can consider End(F[x]). One element of End( F[x]) acts on each polynomial in F[x] by multiplying it by x. Let this endomorphism be X, so  X(a₀ + a₁x + a₂x² + · · · + a_nxⁿ) = a₀x + a₁x₂ + a₂x³ + · · · + a_nxⁿ⁺¹. 

Another element of End(F[x]) is formal differentiation with respect to x. (The familiar formula "the derivation of a sum is the sum of the derivatives" guarantees that differentiation is an endomorphism of F[x].) Let Y be this endomorphism, so Y(a₀ + a₁x + a₂x² + · · · + a_nxⁿ) = a₁ + 2a₂x + · · · + na_nx^n-1.

Exercise 17 asks us to show that Y X - XY = 1, where 1 is unity (the identity map) in End(F[x ]). Thus XY ≠ Y X. Multiplication of polynomials in F[x] by any element of F also gives an element of End (F[x]). The subring of End(F[x]) generated by X and Y and multiplications by elements of F is the Weyl algebra and is important in quantum mechanics.