Let $A = \sum_{k=0}^{p} a_k X^k$ be an element of $\mathbb{K}[X]$, of degree $p \geqslant 0$, which without loss of generality we assume to be monic. In this question, we assume $p \geqslant 1$ and $A = (X - \lambda)^p$, with $\lambda$ in $\mathbb{K}^*$. We denote by $E_A(\mathbb{K})$ the set of $x$ in $\mathbb{K}^{\mathbb{N}}$ with general term $x_n = Q(n)\lambda^n$, where $Q$ is in $\mathbb{K}_{p-1}[X]$. a) Show that $E_A(\mathbb{K})$ is a vector subspace of $\mathbb{K}^{\mathbb{N}}$ and specify its dimension. b) Show the equality $\mathcal{R}_A(\mathbb{K}) = E_A(\mathbb{K})$.
Let $A = \sum_{k=0}^{p} a_k X^k$ be an element of $\mathbb{K}[X]$, of degree $p \geqslant 0$, which without loss of generality we assume to be monic. In this question, we assume $p \geqslant 1$ and $A = (X - \lambda)^p$, with $\lambda$ in $\mathbb{K}^*$.
We denote by $E_A(\mathbb{K})$ the set of $x$ in $\mathbb{K}^{\mathbb{N}}$ with general term $x_n = Q(n)\lambda^n$, where $Q$ is in $\mathbb{K}_{p-1}[X]$.
a) Show that $E_A(\mathbb{K})$ is a vector subspace of $\mathbb{K}^{\mathbb{N}}$ and specify its dimension.
b) Show the equality $\mathcal{R}_A(\mathbb{K}) = E_A(\mathbb{K})$.