2) Let V be an F vector space and let V₁,..., Vn be a subspaces of V. The point of this problem is to show the "internal" and "external" definitions of direct sum are equivalent. That is: prove the following are equivalent (a) V = V₁ ++ Vn and VinΣjti Vj = {Ov} (b) Every vector v eV has a unique expression v = v₁ + ... + Un with v₂ € Vi (c) V = V₁ +. + V₁ and the map T: V₁ V₂ → V₁ + ··· + Vn π(V₁,..., Un) = V₁ + ... + Vn is an isomorphism Can we replace the sum condition in (a) by just asking that Vin V₂ = {0v} for all i + j?

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