1. ## Linearly independent subset

Let V=P(R), and for $\displaystyle j \geq 1$ define $\displaystyle T_{j}(f(x)) = f^{(j)}(x)$, where $\displaystyle T_{j}(f(x)) = f^{(j)}(x)$ is the jth derivative of f(x). Prove that the set $\displaystyle \{ T_{1},T_{2},...,T_{n} \}$ is a linearly independent subset of $\displaystyle \iota (V)$, the vector space of all linear transformations from V into V.

I don't really know how to even start...

2. What is V = P(R)?

3. Sorry, T is the linear transformation from V to V, and V equals to P(R), which is the set of all polynomials with coefficients from a field F.

4. I think I just solved the problem.

I assume to the contrary that the set is linearly dependent for all f(x). Then I pick $\displaystyle f(x) = x^2+x+1$, then $\displaystyle \sum ^{n}_{1}a_{i}T_{i}(f(x))=\sum ^{n}_{1}a_{i}f^{(i)}(x)=0$, but then I found $\displaystyle a_{i}=0 \ \ \forall i$, which is impossible.

Sorry, I think I posted this problem up a bit too hasty, I should have been able to solve it earlier.

Thank you!