This problem provides a linear algebra method for computing sums of the form $\displaystyle S_k=1^k+2^k+3^4+\ldots+n^k$ with $\displaystyle k$ positive integer.

1. Let $\displaystyle \mathbb{R}_5[x]$ the real vector space of all polynomials of degree $\displaystyle \leq 5$. Consider the map $\displaystyle T:\mathbb{R}_5[x]\rightarrow{\mathbb{R}_5[x]}$ defined by $\displaystyle \forall p(x) \in \mathbb{R}_5[x],\;T(p(x))=p(x+1)-p(x)$ . Prove that $\displaystyle T$ is a linear map.

2. Prove that $\displaystyle p\in \ker T\Leftrightarrow\;p$ is constant.

3. Find the image by $\displaystyle T$ of the elements of the canonical basis of $\displaystyle \mathbb{R}_5[x]$ and determine $\displaystyle T^{-1}(x^4)$ .

4. Using **3. ,** find an expression for $\displaystyle S_4=1^4+2^4+3^4+\ldots+n^4$ .