Exercise 2.7.12

Question

Let $V$ be the solution space of an $n^{th}$-order homogeneous linear differential equation with constant coefficients having auxiliary polynomial $p(t)$. Prove that if $p(t) = g(t)h(t)$, where $g(t)$ and $h(t)$ are polynomials of positive degree, then 
$N(MD)) = R(g(Dv)) = p(D)(V)$, 
where $Dv: V \rightarrow V$ is defined by $Dv(x) = x'$ for $x \in V$. Hint: First prove $g(D)(V) \subset N(/i(D))$. Then prove that the two spaces have the same finite dimension.

Jephian C.-H. Lin · Accepted Answer

The second equality is the definition of range. To prove the first equality, we observe that $R(g(D_V))\subset N(h(D))$ since 
$$h(D)(g(D)(V))=p(D)(V)=\{0\}.$$
Next observe that 
$$N(g(D_V))=N(g(D))$$
since $N(g(D))$ is a subspace in $V$. By Theorem 2.32, the dimension of $N(g(D_V))=N(g(D))$ is the degree of $g$. So the dimension of $R(g(D_V))$ is the degree of $h(t)$ minus the degree of $g(t)$, that is the degree of $h(t)$. So $N(h(D))$ and $R(g(D_V))$ have the same dimension. Hence they are the same.

Exercise 2.7.12

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Exercise 2.7.12

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