I am taking a first course on ordinary differential equations(with background in linear algebra). I would like to know, if the proof to the below claim is correct (and that I am not using advanced facts to prove basic facts). Prove that every expression of the form: $$a_n(x)D^n+a_{n-1}(x)D^{n-1}+\ldots+a_1(x)D+a_0(x)$$ defines a linear transformation from $C^n[a,b]$ to […]

- Tags \ldots, $\frac{d^n}{dx^n}(u + v)=\frac{d^n}{dx^n}(u)+\frac{d^n}{dx^n}(v)$ $\frac{d^n}{dx^n}(\alpha u)=\alpha \frac{d^n}{dx^n}(u)$ Therefore, $a_0(x)I, $D^n$ sends a function $f$ in $C^n[a, a_1(x)D, a_2(x)D^2, a_n(x)$ are continuous on $[a, a_n(x)D^n$ are also linear transformations. (3) Since, b]\}$ is a vector space. Vector addition is defined in the usual way functions are added point-wise. $(S+T)(v)=S(v)+T(v)$. Scalar multiplicat, b]\rightarrow C[a, b]$ and $C[a, b]$ are finite-dimensional vector spaces, b]$ to $C[a, b]$ to $h$ in $C[a, b]$ whenever $a_0(x), b]$. (2) From high-school calculus, b]$. Proof. (1) Consider $\frac{d^n}{dx^n}(y) = h$. Any function $y$ that satisfies this equation must have atleast $n$ derivatives. In ot, cna, d, D^n$ are linear maps. In fact, d2, i, I am taking a first course on ordinary differential equations(with background in linear algebra). I would like to know, if $u, if the proof to the below claim is correct (and that I am not using advanced facts to prove basic facts). Prove that every expression of the, the set of all such linear transformations from $\{T:C^n[a, v$ functions $n$ times differentiable