4. Suppose that V and W are vector spaces over a field F and let L :V → W be a function. a) Prove that L is a linear transformation if and only if L(Au + v) = \L(u) + L(v) for all A E F and u, v E V....

Extracted text: 4. Suppose that V and W are vector spaces over a field F and let L :V → W be a function. a) Prove that L is a linear transformation if and only if L(Au + v) = \L(u) + L(v) for all A E F and u, v E V. b) Let v1,..., Vn E V and A1,.….., An E F. If L is a linear transformation, prove that L(A1v1 + ...+AnVn) = A1L(v1)+...+ AnL(Vn). (Hint: induction.)