Suppose $\vect{u}\in\complex{m}$ and $\alpha\in\complex{\null}$, then the scalar multiple of $\vect{u}$ by $\alpha$ is the vector $\alpha\vect{u}$ defined by \begin{align*} \vectorentry{\alpha\vect{u}}{i} &=\alpha\vectorentry{\vect{u}}{i} &&1\leq i\leq m \end{align*}