Prove that a function T: Fn→Fm is linear if and only if there exist f1, f2, . . . , fm ∈ (Fn)∗ such that T(x) = (f1(x), f2(x), . . . , fm(x)) for all x ∈ Fn. Hint: If T is linear, define fi(x) = (giT)(x) for x ∈ Fn; that is, fi= Tt(gi) for 1 ≤i≤m, where {g1, g2, . . . , gm} is the dual basis of the standard ordered basis for Fm.

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