e.g:
Let V be the span of sinx and cosx in C∞, T(f)= 3f + 2f'- f'', T: V → V,
Is T an isomorphism?
How do I solve this kind of questions? I know it has to satisfy: 1. Linear Transformation 2. Invertible But I'm wondering where to start. And I'm really thankful for your help.
Hint: Just use the definitions. If you want know that T is linear, prove that $T(\alpha f+\beta g)=\alpha T(f)+\beta T(g)$ holds for all $\alpha, \beta \in V$. If you want to know that $T$ is invertible, prove that it is both injective ind surjective. (Note: For the finite dimensional vector space case, it is only necessary to prove one of them)
Besides, as $[ \sin, \cos ]$ is a basis of $V$, for all $f\in V$, suppose f = $a\cos+b\sin$. Then $$ \begin{align} T(f) &= 3f+2f'-f'' \\&=3(a\cos+b\sin)+2(-a\sin+b\cos)-(-a\cos-b\sin) \\&=(4a+2b)\cos+(-2a+4b)\sin \end{align} $$ and the coordinate transform is $(a,b)\mapsto(4a+2b, -2a+4b)$. It shouldn't be hard to prove that it is both linear and invertible.