Answer
$$
S=\left\{ \left[\begin{array}{lll}{1}\\ {0} \\ {0} \\{0}\end{array}\right], \left[\begin{array}{lll}{0} \\ {1} \\ {0} \\{0}\end{array}\right], \left[\begin{array}{lll}{0} \\ {0} \\ {1} \\{0}\end{array}\right],\left[\begin{array}{lll}{0} \\ {0} \\ {0} \\{1}\end{array}\right]\right\}.
$$
Work Step by Step
The standard basis of $M_{4,1}$ is given by
$$
S=\left\{ \left[\begin{array}{lll}{1}\\ {0} \\ {0} \\{0}\end{array}\right], \left[\begin{array}{lll}{0} \\ {1} \\ {0} \\{0}\end{array}\right], \left[\begin{array}{lll}{0} \\ {0} \\ {1} \\{0}\end{array}\right],\left[\begin{array}{lll}{0} \\ {0} \\ {0} \\{1}\end{array}\right]\right\}.
$$
