In this paper, we characterize the extremal dependence of $d$ asymptotically dependent variables by a class of random vectors on the $(d-1)$-dimensional hyperplane perpendicular to the diagonal vector $\mathbf1=(1,\ldots,1)$. This translates analyses of multivariate extremes to that on a linear vector space, opening up possibilities for applying existing statistical techniques that are based on linear operations. As an example, we demonstrate obtaining lower-dimensional approximations of the tail dependence through principal component analysis. Additionally, we show that the widely used H\"usler-Reiss family is characterized by a Gaussian family residing on the hyperplane.

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