Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$, $p \in \Sigma_{N}$, $m \in \mathbb{R}^{d}$, and $A \in \mathscr{M}_{d}(\mathbb{R})$ defined by $A_{lk} = \sum_{i=1}^{N} p_{i}(M_{il} - m_{l})(M_{ik} - m_{k})$. We denote by $\widetilde{M} = (M \mid \mathbf{1}) \in \mathscr{M}_{N,d+1}(\mathbb{R})$ the augmented matrix obtained by adding a column of 1s to the right of $M$. Let $\theta \in \mathbb{R}^{d}$ such that $\theta^{T} A \theta = 0$. We assume that $p_{i} \neq 0$ for all $1 \leqslant i \leqslant N$. (a) Show that there exists $c \in \mathbb{R}$, which you will specify, such that for all $i \in \{1, \ldots, N\}$, we have $\sum_{l=1}^{d} M_{il} \theta_{l} = c$. (b) Show that if $\ker \widetilde{M} = \{0\}$ then $\theta = 0$.
Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$, $p \in \Sigma_{N}$, $m \in \mathbb{R}^{d}$, and $A \in \mathscr{M}_{d}(\mathbb{R})$ defined by $A_{lk} = \sum_{i=1}^{N} p_{i}(M_{il} - m_{l})(M_{ik} - m_{k})$. We denote by $\widetilde{M} = (M \mid \mathbf{1}) \in \mathscr{M}_{N,d+1}(\mathbb{R})$ the augmented matrix obtained by adding a column of 1s to the right of $M$.
Let $\theta \in \mathbb{R}^{d}$ such that $\theta^{T} A \theta = 0$. We assume that $p_{i} \neq 0$ for all $1 \leqslant i \leqslant N$.
(a) Show that there exists $c \in \mathbb{R}$, which you will specify, such that for all $i \in \{1, \ldots, N\}$, we have $\sum_{l=1}^{d} M_{il} \theta_{l} = c$.
(b) Show that if $\ker \widetilde{M} = \{0\}$ then $\theta = 0$.