Consider a simplex $T$ in $R^d$ with $N_1(T) = \left\{N_i\right\}_{i=0}^{d}\subset P_1^{*}(T)$ be the Lagrange nodal variables (or nodal evaluation). By the Riesz representation theorem, there exist functions $\lambda_{i}^{*}\in P_1(T)$ (which is the representation of $N_i\in P_1(T)$) such that $N_j(\lambda_{i}) = \int_{T} \lambda_{i}\lambda_{j}^{*} = \delta_{ij}$ for each $0\leq i\leq d$. Show that $\lambda_{i}^{*} = \frac{(1+d)^2}{|T|}\lambda_{i} - \frac{1+d}{|T|}\sum_{j\neq i} \lambda_j$

My attempt: I was thinking that $\lambda_i^{*}$ plays a role like a Fourier coefficient, so I multiply both sides by $\lambda_j$ and integrate over $T$. Then LHS would be $\delta_{ji}$, while RHS $= \frac{(1+d)^2}{|T|}\int_{T} \lambda_{i}\lambda_{j} - \frac{1+d}{|T|}\sum_{j\neq i}\int_{T} \lambda_{i}\lambda_{j}$. Now, for $i=j$, then $LHS=1$, and I was thinking of $\int_{T} \lambda_{i}\lambda_{i} = $ the area of region $T$, but then if that's the case, the RHS would be $(1+d)^2,$ which is certainly not equal to $1$ (contradiction). So I don't think my interpretation is correct, but I could not see how to proceed further without knowing the explicit formula for $\lambda_i$.

My question: Could anyone please help me with this problem? It seems to be such a beautiful identity, yet quite hard to prove. Any thoughts would really be appreciated.


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