Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Please help with the following proof:

Let $f(x)$ be a polynomial of degree n in $P_n(\mathbb{R})$. Prove that for any $g(x)$ element of $P_n(\mathbb{R})$ there exist scalers $c_0,c_1,\ldots,c_n$ such that
$$g(x) = c_0f(x) + c_1f'(x) + c_2f''(x) + \cdots + c_nf^{(n)}(x)$$

share|improve this question
1  
Try to prove that the set $\{ f(x),f'(x),\ldots,f^{(n)}(x) \}$ is linearly independent. –  user2468 Feb 9 '12 at 17:14
add comment

2 Answers

Hint: If you have polynomials $p_0(x)$, $p_1(x),\ldots,p_n(x)$ with real coefficients, with $p_i(x)$ of degree $i$, are they linearly independent?

share|improve this answer
add comment

Hint: the coefficients of powers of $x$ in your equation give you $n+1$ equations in the $n+1$ unknowns $c_0, \ldots, c_n$. The coefficient matrix for this system is triangular with nonzero elements on the diagonal.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.