\displaystyle $\begin{align*}
& r([a_1, \ldots, a_k, b]) = r([a_1, \ldots, a_k]) \\
\Leftrightarrow \ & \dim[\{ a_1, \ldots, a_k, b \}] = \dim[\{ a_1, \ldots, a_k \}] \\
\Leftrightarrow \ & [\{ a_1, \ldots, a_k, b \}] = [\{ a_1, \ldots, a_k \}] \\
\Leftrightarrow \ & b \in [\{ a_1, \ldots, a_k \}] \\
\Leftrightarrow \ & (\exists \alpha_1, \ldots, \alpha_k \in \mathbb{F}) \ b = \sum_{i = 1}^k \alpha_i a_i
\end{align*}$