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Math Help - Linearly independent subset

  1. #1
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    Linearly independent subset

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

    I don't really know how to even start...
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  2. #2
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    What is V = P(R)?
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  3. #3
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    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.
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  4. #4
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    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 f(x) = x^2+x+1, then \sum ^{n}_{1}a_{i}T_{i}(f(x))=\sum ^{n}_{1}a_{i}f^{(i)}(x)=0 , but then I found  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!
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