# Linear algebra proof involving polynomials.

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
$$g(x) = c_0f(x) + c_1f'(x) + c_2f''(x) + \cdots + c_nf^{(n)}(x)$$

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Try to prove that the set $\{ f(x),f'(x),\ldots,f^{(n)}(x) \}$ is linearly independent. –  user2468 Feb 9 '12 at 17:14
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.
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?