The answer is “True”.

The span of $\{\left[\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 1\\ 0\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 0\\ 1\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right]\}$ with real numbers as scalars is ${\mathbb{R}}^4$.