If \(V\) is a vector space over the field \(F\), verify that \[ (\alpha_1 + \alpha_2) + (\alpha_3 + \alpha_4) = [\alpha_2 + (\alpha_3 + \alpha_1)] + \alpha_4 \] for all vectors \(\alpha_1, \alpha_2, \alpha_3\), and \(\alpha_4\) in \(V\).
Solution
Using associativity and commutativity of vector addition, \[ (\alpha_1+\alpha_2)+(\alpha_3+\alpha_4) =\alpha_1+\alpha_2+\alpha_3+\alpha_4. \] Also \[ [\alpha_2+(\alpha_3+\alpha_1)]+\alpha_4 =\alpha_2+\alpha_3+\alpha_1+\alpha_4 =\alpha_1+\alpha_2+\alpha_3+\alpha_4. \] Thus the two sides are equal.