Consider a vector function $V_{n\times 1}(m)$, where $m\in \mathbb{N}$, has the property that:
$V_{n\times 1}(1)=\left[ \begin{matrix} 1 \\ 0 \\ \vdots \\ 0 \\ \end{matrix} \right]$
$V_{n\times 1}(2)=\left[ \begin{matrix} 0 \\ 1 \\ 0 \\ \vdots \\ \end{matrix} \right]$
and so on.
could you design polynomials for enteries in terms of $m, n$ that leads to above vector function.
It's very nice of you to help me.
Consider the following polynomials :$$ p_1(m)= \widehat{(m-1)}(m-2)\ldots(m-n)$$ $$ p_2(m)=(m-1)\widehat{(m-2)}\ldots(m-n)$$ $$\vdots $$ $$ p_n(m)= (m-1)(m-2)\ldots\widehat{(m-n)}$$ where the hat signifies omission of the factor. Your vector now looks like: $V_{n \times 1}(m)=[p_1(m)/p_1(1),p_2(m)/p_2(2),\ldots,p_i(m)/p_i(i),\ldots,p_n(m)/p_n(n)]^T$.