Let T : V → V be the adjacency operator of the Petersen graph as illustrated in the enclosed file. Here V is the vector space of all formal real linear combinations of the vertices v1, ... , U10 of the Petersen graph. 1. Show that the subspace spanned by u = v1 + v2 + V3 + V4 + v5 and w = v + v7 + Vg + v9 + V10 is stable under T, by calculating T(u) and T(w) explicitly. 2. Use part 1 to calculate (by hand) an eigenvector for T satisfying T(v) = 3v, and an eigenvector satisfying T(v) = v.

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