On page 9, Parrilo wrote$^\color{magenta}{\star}$ that the vector representation of a spectrahedron
$$ S = \left\{ (x_1, \dots, x_m) \in {\Bbb R}^m : A_0 + \sum_{i=1}^m A_i x_i \succeq 0 \right\} $$
is affine equivalent to the set representation of a spectahedron
$$ \left\{ A_0 + \sum_{i=1}^m A_i x_i \mid x \in \mathbb{R}^m \right\} \bigcap \mathcal{S}_+^n $$
Affine equivalent, as far as I understand, means we can produce a bijective linear transformation from set $A$ to set $B$. What function maps from the set of matrices of a spectrahedron to the associated set of vectors?
$\color{magenta}{\star}$ Semidefinite Optimization and Convex Algebraic Geometry, SIAM, December 2012.