Show-that-n-N-u-0-u-1-u-n-v-0-v-1-v-n-C-k-n-u-k-i-0-k-k-i-v-i-k-n-v-k-i-0-k-1-k-1-k-i-u-i-

Question Number 115218 by Ar Brandon last updated on 24/Sep/20

Show that ∀n∈N, ∀u_0 ,u_1 ,...,u_n ,v_0 ,v_1 ,...v_n ∈C ∀k≤n; u_k =Σ_(i=0) ^k ((k),(i) )v_i ⇔∀k≤n; v_k =Σ_(i=0) ^k (−1)^(k−1) ((k),(i) )u_i

$$\mathrm{Show}\:\mathrm{that}\:\forall\mathrm{n}\in\mathbb{N},\:\forall\mathrm{u}_{\mathrm{0}} ,\mathrm{u}_{\mathrm{1}} ,…,\mathrm{u}_{\mathrm{n}} ,\mathrm{v}_{\mathrm{0}} ,\mathrm{v}_{\mathrm{1}} ,…\mathrm{v}_{\mathrm{n}} \in\mathbb{C} \\ $$$$\forall\mathrm{k}\leqslant\mathrm{n};\:\mathrm{u}_{\mathrm{k}} =\sum_{\mathrm{i}=\mathrm{0}} ^{\mathrm{k}} \begin{pmatrix}{\mathrm{k}}\\{\mathrm{i}}\end{pmatrix}\mathrm{v}_{\mathrm{i}} \Leftrightarrow\forall\mathrm{k}\leqslant\mathrm{n};\:\mathrm{v}_{\mathrm{k}} =\sum_{\mathrm{i}=\mathrm{0}} ^{\mathrm{k}} \left(−\mathrm{1}\right)^{\mathrm{k}−\mathrm{1}} \begin{pmatrix}{\mathrm{k}}\\{\mathrm{i}}\end{pmatrix}\mathrm{u}_{\mathrm{i}} \\ $$

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