Let p be an odd prime (i.e., any prime other than 2). Form two vector spaces...

Let p be an odd prime (i.e., any prime other than 2). Form two vector spaces V₁, V₂ over F_p (prime field of order p) with bases corresponding to the edges and faces of an icosahedron (so that V₁ has dimension 30 and V2 has dimension 20). Let T : V₁ → V₂ be the linear transformation defined as follows: given a vector v ∈ V₁, T(v) is the vector in V₂ whose component corresponding to a given face is the sum of the components of v corresponding to the edges around that face. Prove that T is surjective. (Hint: one option is to look closely at the five edges emanating from a single vertex.)

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Colby Messinger answered 1 year ago

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