T: P2(R)--->P3(R) defined by T(F(x))=xf(x)+f'(x).
I need to prove that T is a linear transformation, and say if is one 2 one or onto.
Hello,
Note that the "object" of T is the function f, not x.
Knowing that, you have to prove, in order to say that T is a linear transformation :
$\displaystyle T(aF(x))=aT(F(x))$, $\displaystyle \forall a \in \mathbb{R}^*$
$\displaystyle T(F(x)+G(x))=T(F(x))+T(G(x))$
Or you can just say that T is the sum of linear transformations :
- multiplying by a scalar : x
- taking the derivative of a function, though you have to say that it's differentiable..