Consider the summation:
\begin{align} \sum_{i=1}^n \sum_{j=1}^k a_ib_j = \ & a_1 b_1 + a_1 b_2 + \cdots + a_1 b_k + \\ & a_2 b_1 + a_2 b_2 +\cdots + a_2 b_k + \\ & \quad \vdots \\ & a_n b_1 + a_n b_2 + \cdots + a_n b_k \end{align}
Since addition is commutative, we can group terms that share the same \(b_j\):
\begin{align} \sum_{i=1}^n \sum_{j=1}^k a_ib_j = \ & a_1 b_1 + a_2 b_1 + \cdots + a_n b_1 + \\ & a_1 b_2 + a_2 b_2 + \cdots + a_n b_2 + \\ & \quad \vdots \\ & a_1 b_k + a_2 b_k + \cdots + a_n b_k \\ = & \sum_{j=1}^k\sum_{i=1}^n a_ib_j\end{align}