# Thread: Invertibility of a Function

1. ## Invertibility of a Function

This question Im really lost on, a bit of guidance would be great!

Let $J_n$ be the $R$-vector space with basis $B = \{1,cos(x),sin(x),...,cos(nx),sin(nx)\}$. For a fixed positive real number $a$, define $D \in Hom(J_n,J_n)$ by $D(f) = f'' + a^2f$. For which $a$ is $D$ invertible?

2. Originally Posted by joe909
This question Im really lost on, a bit of guidance would be great!

Let $J_n$ be the $R$-vector space with basis $B = \{1,cos(x),sin(x),...,cos(nx),sin(nx)\}$. For a fixed positive real number $a$, define $D \in Hom(J_n,J_n)$ by $D(f) = f'' + a^2f$. For which $a$ is $D$ invertible?

Check the efect of the transformation on the given basis' elements:

$D(1)=a^2\Longrightarrow$ so far it must be $a\neq 0$

$D(\cos x)=-\cos x +a^2\cos x=\cos x(a^2-1)\Longrightarrow$ it also has to be $a\neq \pm 1$

$D(\cos 2x)=-4\cos x+a^2\cos x =\cos 2x(a^2-4)\Longrightarrow$ it also has to be $a\neq \pm 2$ ....

Tonio

3. Thanks, I get it, if $D(x) = 0$ for $x \neq 0$ then it is no longer one to one. I think I just got caught up in wording of the question.